Method for the structural analysis of panels consisting of an isotropic material and stiffened by triangular pockets

ABSTRACT

A method for dimensioning of panels stiffened by triangular pockets that make it possible to take into account aeronautical specifics and more particularly the stresses that are admissible for the different types of buckling and the calculations of adapted reserve factors is presented. The method relates to dimensioning of a substantially plane panel of homogeneous and isotropic material by an analytical procedure, wherein the panel is composed of a skin reinforced by an assembly of three parallel bundles of stiffeners integrated with the panel, and triangular pockets defined on the skin, the stiffeners are strip-shaped and the panel must satisfy a specification of mechanical resistance to predetermined external loads, including steps organized in such a way that they can be repeated iteratively for different values of input data until reserve factors are obtained to determine the dimensions and arrangement of the panel elements necessary to obtain the imposed mechanical resistance.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a divisional of U.S. application Ser. No.13/395,744, filed May 25, 2012, which is the National Stage ofPCT/FR2010/051900, filed Sep. 13, 2010, which claims the benefit ofpriority to French Application No. 0956286, filed Sep. 14, 2009, theentire contents of each of which is incorporated herein by reference.

FIELD OF THE INVENTION

The present invention is from the domain of structures. It particularlyconcerns structures of a stiffened panel type, and even moreparticularly such panels which are reinforced by stiffeners. Theinvention is concerned with calculating the resistance of suchstructures subjected to combined loads.

PRIOR ART AND THE PROBLEM POSED

Thin, stiffened structures represent the greater part of primarycommercial aircraft structures.

Panels are generally reinforced with stiffeners which are perpendicularto each other and which define rectangular zones on the skin of thepanel, limited by stiffeners and referred to as pockets.

The structure of an aeroplane is thus conceived with a skeleton ofstiffeners and a skin:

-   -   longitudinal stiffeners (generally referred to as longerons):        they provide support for the structure in the principal        direction of loads    -   transversal stiffeners (generally called        frame” or “rib”): their main function is to provide support for        the longerons    -   a panel (generally called skin): as a general rule, it takes up        loading in the plane (membrane)

The longerons and the stringers are set at 90° to each other and definerectangular pockets on the skin.

However, during the 1950's and 60's, for spacecraft structures, NASAdeveloped a new concept for stiffened structures called “Isogrid” (seeFIG. 1).

Such a stiffened structure thus composed of a reinforced skin with anetwork of stiffeners set at θ° (θ=60°, in the structures envisaged byNASA) between them. The stiffeners are blade shaped and are built intothe panel. Because of its geometry, this configuration possessesorthotropic qualities (isotropic when θ=60°) and the pockets formed onthe skin are triangular.

In the following description, the terms structure stiffened bytriangular pockets or panel stiffened by triangular pockets are used todefine the structures or panels reinforced by crossed stiffeners formingtriangular pockets.

Limited data is available in literature for calculating the resistanceand the stability of such a structure stiffened by triangular pockets.

State of the Art of Calculation Methods for Panels Stiffened byTriangular Pockets

A method for the analytical calculation of panels stiffened byequilateral triangular pockets is described in the NASA Contract Report“Isogrid” design handbook” (NASA-CR-124075, 02/1973)

This method is well documented, but presents some serious limitations:use of equilateral triangles only: angle=60°, calculation of appliedstresses but no calculation of stress capacity, Poisson coefficient ofthe material equal to only ⅓.

The prior method presents many limitations and does not take intoaccount all the problems which are presented on an aircraft structure,in particular concerning boundary conditions and plasticity. It cannottherefore be used reliably for the analytical calculation of thestructure of panels stiffened by triangular pockets.

OBJECTIVES OF THE INVENTION

In order to carry out a structural analysis of panels stiffened bytriangular pockets, a method for structural analysis was developed,based on a theory of composite plate and taking into account itsspecific modes of failure. This method is applied to flat panels made ofa material with isotropic properties.

The method described herein envisages a modification of the base anglebetween the stiffeners (which is 60° in the “Isogrid” structures). Thissignifies that the isotropic quality of the panel is no longerguaranteed.

EXPLANATION OF THE INVENTION

The invention relates, to this effect, to a method of dimensioning by ananalytical method, an essentially flat panel consisting of a homogenousand isotropic material, the panel being composed of a skin reinforced bya set (known as “grid”) of three parallel bundles of stiffeners builtinto the panel, the pockets determined on the skin by said groups ofstiffeners are triangular, the stiffeners are blade shaped and thestiffened panel must comply with specifications for mechanicalresistance to predetermined external loads, the angles between bundlesof stiffeners being such that the triangular pockets have any kind ofisosceles shape.

According to one advantageous implementation, the method includes thesteps:

Step 2 of calculating the stresses applied in the skin and thestiffeners, as well as the flow in the skin and loads in the stiffeners,based on the geometry of the stiffened panel, and the external loads,assumed to be in the plane of the panel and applied at the centre ofgravity of the section (of the panel), the stiffened panel beingrepresented by an assembly of two orthotropic plates, the grid ofstiffeners being represented by an equivalent panel.

Step 3—of calculating the internal loads of the stiffened panel,

Step 4—of resistance analysis including a calculation of reserve factorsof the material at capacity and ultimate load,

Step 5—of calculating the local stress capacity,

In preference, the method takes into account the redistribution ofapplied stresses between the panel and the grid of stiffeners due:

to the post-buckling of stiffeners, by the definition of an effectivestraight section for each type of stiffener (0°, +θ or −θ). A_(0°)^(st), A_(+θ) ^(st) and A_(−θ) ^(st),

to the post-buckling of the pocket through the calculation of aneffective thickness of the panel: t_(s) _(_) _(eff),

to the plasticity of applied external loads, through an iterativeprocess on the various properties of the material, in particular Young'smodulus and Poisson coefficients: E_(0°) ^(st), E_(+θ) ^(st), E_(−θ)^(st) for the stiffeners and E_(x) ^(s), E_(y) ^(s) and ν_(ep) ^(st) forthe skin, using the Ramberg-Osgood law.

According to a preferred mode of implementation of the method accordingto the invention, this includes a step of correcting the applied loadsto take into account plasticity, using an iterative method forcalculating the plastic stresses, carried out until the five parametersof the material (E_(0°) ^(st), E_(+θ) ^(st), E_(−θ) ^(st), E_(skin),ν_(ep)) entered at the start of the process are noticeably equal to thesame parameters obtained after the calculation of plastic stress.

According to an advantageous implementation, the method includes a step4, of analysing resistance comprising a calculation of the reservefactors of the material at a capacity and ultimate load, carried out bycomparing the applied loads calculated in the stiffened panel componentswith the maximum stress capacity of the material, the applied loadsbeing corrected to take into account the plasticity of the stiffenedpanel.

According to an advantageous implementation, the method includes a step5 for calculating the local stress capacity, which includes a sub-step5A of calculating the buckling flow capacity, and the reserve factor forthe isosceles triangular pockets, the applied stresses to be taken intoaccount for the calculation of the reserve factor being only thestresses affecting the skin, the external flows used being the flows ofthe skin do not correspond to the stiffened panel being fully loaded.

In this case, step 5A of calculating the buckling flow capacity andreserve factor for the isosceles triangular pockets favourably includestwo sub-steps: firstly of calculating the capacity values for platessubjected to cases of pure loading (compression according to twodirections in the plane, shear load) by using a finite element method,then calculating the interaction curves between these cases of pureloading.

Even more precisely, calculating the capacity values includes thefollowing sub-steps of:

-   -   Creating an FEM parametric model of a triangular plate    -   Testing various combinations to obtain buckling results,    -   Obtaining parameters that are compatible with an analytical        polynomial formula

In a particular mode of implementation, in the case of pure loading, theinteraction curves are defined by the following sub-steps:

-   -   of creating finite element models of several triangular plates        with different isosceles angles, the isosceles angle (0) being        defined as the base angle of the isosceles triangle,    -   for each isosceles angle:        -   1/ of calculating by Finite Element Model to determine the            flow capacity of wrinkling (without plastic correction) for            various plate thicknesses.        -   2/ of tracing a flow curve of buckling capacity according to            the

$\frac{D}{h^{2}}$

-   -   -    ratio (D plate stiffness, h height of the triangle), this            curve being determined for the small values of

$\frac{D}{h^{2}},$

-   -   -    by a second degree equation according to this ratio, of            which the coefficients K₁ and K₂ depend on the angle and the            load case being considered,        -   3/ of tracing the evolution of coefficients of the            polynomial equation K₁ and K₂ according to the base angle of            the isosceles triangle, these coefficients being traced            according to the angle of the triangular plates being            considered, and interpolation to determine a polynomial            equation which makes it possible to calculate the constants            whatever the isosceles angle.

Again, in the case of calculation of the buckling flow capacity andreserve factor of isosceles triangular pockets, according to anadvantageous implementation, in the case of combined loading, thefollowing hypothesis is used: if some components of the combined loadare under pressure, these components are not taken into account for thecalculation, and the interaction curves are defined by the followingsub-steps:

-   -   of creating finite element models of several triangular plates        with different isosceles angles, the isosceles angle (0) being        defined as the base angle of the isosceles triangle,    -   for each angle,        -   1/ of calculating by Finite Element Model (FEM) to determine            the eigenvalue of buckling that corresponds to the various            distributions of external loads.        -   2/ of tracing the interaction curves, for each angle and            each combination of loads and approximating these curves            with a unique equation covering all these combinations:

${R_{cX}^{A} + R_{cy}^{B} + R_{s}^{C}} = {1\mspace{14mu} \left( {{{{or}\mspace{14mu} R_{i}} = \frac{N_{i}^{app}}{N_{i}^{crit}}},} \right.}$

equations in which R_(i) represents the load rate and N_(i) ^(app) andN_(i) ^(crit) the applied flows and critical flows for i=cX, cY or s,corresponding to cases of compression according to axes X and Y, and toa case of shear load), A, B, C being empirical coefficients.

Advantageously, the method also comprises a sub-step of calculatingreserve factors, by solving the following equation:

${{\left( \frac{R_{cY}}{R} \right)^{A} + \left( \frac{R_{cX}}{R} \right)^{B} + \left( \frac{R_{s}}{R} \right)^{C}} = 1}\mspace{11mu}$${{with}\mspace{14mu} R} = {\frac{N_{cY}^{app}}{N_{cYcomb}^{crit}} = {\frac{N_{cX}^{app}}{N_{cXcomb}^{crit}} = {\frac{N_{s}^{app}}{N_{scomb}^{crit}} = \frac{1}{RF}}}}$

According to an advantageous implementation, the method uses, for thecalculation of plasticity corrected stress capacity, a plasticitycorrection factor η, defined by:

-   -   for all cases of loading (pure and combined) with the exception        of shear load,

$\eta_{5} = \frac{E_{\tan}}{E_{c}}$

-   -   for cases of pure shear loading:

$\eta_{6} = {\frac{\left( {1 + v_{e}} \right)}{\left( {1 + v} \right)} \cdot \frac{E_{\sec}}{E_{c}}}$

the plasticity correction being calculated by using the equivalentelastic stress of Von Mises.

According to an advantageous implementation, in the case of simplysupported or clamped isosceles triangular plates, in a case of combinedloading, an interaction curve is used: R_(cX)+R_(cY)+R_(s) ^(3/2)=1, forall cases of loading.

According to an advantageous implementation, the method includes a step5 for calculating the local stress capacity, which includes a sub-step5B of calculating the buckling stress capacity, and reserve factor forthe stiffener web, considered as a rectangular panel, the stressesapplied for calculations of reserve factor being only the stresses inthe stiffener webs.

According to an advantageous implementation of the method, this includesa step 6, of calculating general instability, providing data on bucklingflow capacity, and reserve factors, for a flat stiffened panel, underpure or combined loading conditions, the flows applied, to be taken intoaccount for the calculation of reserve factor being the external flowsof the stiffened panel.

In this case, more specifically, the method advantageously includes thefollowing sub-steps:

-   -   of using a general behaviour law (equation 6-8), defining the        flows and moments relations between flow and moments, on one        hand, and strains, on the other, a state of plane stresses being        considered,    -   using general balance equations (equations 6-9 and 6-10) of an        element of the stiffened panel, linking the flows, moments and        the density of surface strengths,    -   of solving a general differential equation (equation 6-17)        between the stress flows, the surface strength density, strains        and bending stiffeners.

According to a favourable implementation, the method includes aniteration step, making it possible to modify the values of appliedstresses, or the dimensional values of panels, according to the resultsof at least one of steps 3 to 6.

In another respect, the invention relates to a computer programmeproduct including a series of instructions adapted to implement a methodsuch as explained, where this set of instructions is executed on acomputer.

BRIEF DESCRIPTION OF FIGURES

The description which follows, given purely as an example of anembodiment of the invention, is given in reference to the annexedfigures which represent:

FIG. 1—An example of a flat panel stiffened by triangular pockets,

FIG. 2—A definition of loading and of the system of coordinates,

FIG. 3—A geometric definition of a panel

FIG. 4—A junction in a structure stiffened by triangular pockets,

FIG. 5—An example of general instability of a panel stiffened bytriangular pockets,

FIG. 6—The theory of effective width,

FIG. 7—A general organogram of the method according to the invention,

FIG. 8—A decomposition of the grid in elementary triangles,

FIG. 9—An elementary isosceles triangle used in the calculation of thepanel mass,

FIG. 10—An elementary rectangular triangle used in the calculation ofthe panel mass,

FIG. 11—An elementary shape of a stiffener grid in a panel stiffened bytriangular pockets,

FIG. 12—A case of pure loadings of the stiffened plate,

FIG. 13—A diagram of loads on a stiffener,

FIG. 14—An expression of the Kc coefficients according to cases ofboundary conditions,

FIG. 15—A panel of stiffeners considered as an assembly of twoorthotropic plates,

FIG. 16—The loads on an elementary shape of a stiffener grid for a panelstiffened by triangular pockets,

FIG. 17—A method for calculating plasticity corrected applied loads,

FIG. 18—The notation conventions of the elementary isosceles triangle,

FIG. 19—A linear or quadratic interpolation of the K coefficient,

FIG. 20—A case of combined loading,

FIG. 21—Conventions of flow and moments,

FIG. 22—The value of h(α) according to the various boundary conditions,for a case of compression,

FIG. 23—The shear buckling coefficient for a four-sided simply supportedconfiguration,

FIG. 24—A table of shear buckling coefficient values,

FIG. 25—The shear buckling coefficient for a clamped four-sidedconfiguration,

FIG. 26—The evolution of the K1 constant according to the isoscelesangle for a simply supported triangular plate,

FIG. 27—The evolution of the K2 constant according to the isoscelesangle for a simply supported triangular plate,

FIG. 28—The evolution of the K1 constant according to the isoscelesangle for a clamped triangular plate,

FIG. 29—The evolution of the K2 constant according to the isoscelesangle for a clamped triangular plate,

DETAILED DESCRIPTION OF A MODE OF EMBODIMENT OF THE INVENTION

The method for resistance analysis of a metal panel stiffened bytriangular pockets, principally plane, described is intended to beimplemented in the form of a programme on a computer of a known type.

The method is intended to be used for a structure which is principallyplane (stiffeners and skin). The method here-described appliesexclusively to the calculation of typical structural settings with thefollowing boundaries:

-   -   The edges of the studied zone do not border an opening.    -   None of the stiffeners extend outside of the zone studied.    -   Each cross section must be bordered by stiffeners.    -   All the triangular pockets in the skin are assumed to have the        same thickness.    -   All the stiffeners are assumed to have the same dimensions.

This method is used for calculating panels built from a homogenous andisotropic material (for example—but not limited to—metal) for which thedescribing monotonically increasing curves (σ, ε) can be idealised bythe means of formulas such as R&O (see further on).

The simplified organogram of the method according to the invention isillustrated by FIG. 7.

Two types of failure (the occurrence of which is evaluated in steps 4and 6 of the method) can occur on a structure stiffened by triangularpockets: A fault in material (which is the object of step 4): theapplied stresses have reached the maximum stress capacity of thematerial (F_(tu) or F_(su)), global failure: generalised buckling(including the grid of stiffeners) occurs on the whole panel (thisverification is the object of step 6).

In addition, two types of instability (object of step 5) weaken theglobal rigidity of the structure stiffened by triangular pockets but donot cause the global failure of the complete structure:

-   -   Instability of the panel: buckling of the triangular pockets    -   Instability of the stiffeners: buckling of stiffener webs

The buckled sections can only support a part of the load which theycould support before they were buckled. Because of this, the appliedloads are redistributed in the structure.

It is noted that in the present invention, post-buckling calculationsare not processed. Because of this, the two types of buckling referredto above are considered as modes of failure.

Notation and Units

The conventions of notations and axis systems are explained in FIG. 2

A local system of coordinates is defined for each stiffener. An X axisis defined in the plane of the straight section of the stiffener, thisis the exit axis, in the direction of the principal dimension of thestiffener. A Z axis is defined as the normal axis in the plane of theskin, in the direction of the stiffener. Finally, a Y axis is the thirdaxis in a system of straight coordinates.

For forces and loads, a negative sign on a force according to the X axissignifies a compression of the stiffener, a positive sign signifies atension.

A positive bending moment causes a compression in the skin and a tensionin the stiffeners.

The general notations used are defined in the following table.

Symbol Unit Description A mm² Surface I mm⁴ Inertia J mm⁴ Torsionconstant K N/mm Normal rigidity (tension/compression) of a plate D N ·mm Bending rigidity of a plate σ N/mm² Stress ε — Strain κ — Strainoutside of the plane η — Plastic correction factor z mm Coordinates onthe z axis k — Buckling coefficient

Suffixes:

Symbol Unit Description g — grid st — stiffener s — skin

Geometric Characteristics

The geometric characteristics of a panel, considered here as anon-limitative example, are given in FIG. 3.

For the rest of the description, several hypotheses are used. It issupposed that the Z axis is a plane of symmetry for the straight sectionof the stiffener. Also, the dimensions a and h are defined according tothe neutral fibre of a stiffener. In addition, the envisaged panelstiffened by triangular pockets does not have stiffeners on the twosides defined by: X=0 and X=Lx

Symbol Unit Description L_(X) mm length L_(Y) mm width a mm Length ofthe side of a triangle θ ° Angle of the triangle h mm${Height}\mspace{14mu} {of}\mspace{14mu} a\mspace{14mu} {triangle}\mspace{14mu} \left( {= {{\frac{a}{2} \cdot \tan}\mspace{14mu} \theta}} \right)$t mm Thickness of the skin d mm Height of the stiffener web b mmThickness of the stiffener web R_(n) mm Fillet node radius R_(f) mmPocket radius A_(x°) ^(st) mm² Straight section of the stiffeneraccording to axis x v_(p) mm Panel offset between its centre of gravityand the origin point of the local coordinates system v_(w) mm Stiffenernetwork offset between its centre of gravity and the origin point of thelocal coordinates system

Materials

Symbol Unit Description F_(cy) MPa Elastical capacity of the materialunder compression F_(tu) MPa Ultimate tension resistance of the materialF_(su) MPa Ultimate shear resistance of the material σ_(n) MPa Stressreference ε_(ult) — Ultimate plastic strain (=e %) ν_(e) — ElasticPoisson coefficient ν_(p) — Plastic Poisson coefficient (=0.5) ν —Elasto-plastic Poisson coefficient E_(c) MPa Young's elastic modulus incompression E MPa Young's elastic modulus under tension E_(sec) MPaSecant modulus E_(tan) MPa Tangent modulus n_(ec) — Ramberg and Osgood(R&O) coefficient in compression G MPa Shear modulus G_(sec) MPa Secantshear modulus ρ kg/mm³ Material density

Stresses

Symbol Unit Description P_(i) N Normal load with i equal to: 0° for theload applied on a stiffener at 0° x° for the load applied on a stiffenerat x° N N/mm Flow M N · mm Bending moment τ MPa Shear stress σ MPaNormal stress σ_(crit) ^(i) MPa Buckling stress of a panel i incompression τ_(crit) ^(i) MPa Buckling stress of a panel i in shear loadσ_(app) ^(i) MPa Applied stresses on the element i RF — Reserve factorR_(c) — Compression load rate R_(s) — Shear load rate R_(p) — Load ratedue to pressure LL Load limit UL Ultimate load

Definitions

For the rest of the description the following terms are defined.

In a structure stiffened by triangular pockets, grid refers to thecomplete network of single stiffeners.

The term node is used to describe an intersection of several stiffenersin a structure stiffened by triangular pockets (see FIG. 4). Inpractice, it is an element of the complex design including bending radiiin both directions.

When a structure (subject to loads only in its plane) sufferssignificant, visible transversal displacements of loads in the plane, itis said to buckle. FIG. 5a illustrates such a case of local instabilityof a panel stiffened by triangular pockets.

The buckling phenomenon can be demonstrated by pressing the oppositesides of a flat cardboard sheet towards each other. For small loads, thebuckle is elastic (reversible) because the buckle disappears when theload is removed.

Local buckling (or local instability) of plates or shells is indicatedby the appearance of lumps, undulations or waves and is common in plateswhich compose thin structures. When considering stiffened panels, localbuckling, as opposed to general buckling, describes an instability inwhich the panel between the longerons (stiffeners) buckles, but thestiffeners continue to support the panels and do not show anysignificant strains outside of the plane.

The structure can therefore present two states of balance:

-   -   Stable: in this case, displacements increase in a controlled        manner when the loads increase, that is to say that the capacity        of the structure to support additional loads is maintained, or    -   Unstable: in this case, strains instantly increase and the        capacity to support loads rapidly declines

A neutral balance is also possible in theory during buckling, this stateis characterised by an increase in strain without modifying the load

If buckling strains become too great, the structure fails. If acomponent or a part of a component is likely to suffer buckling, thenits conception must comply with the stresses of both resistance andbuckling.

General instability refers to the phenomenon which appears when thestiffeners are no longer able to counteract the displacements of thepanel outside of the plane during buckling.

FIG. 5b shows an example of global buckling in compression of astructure stiffened by triangular pockets, when the panel reaches itsfirst mode of general buckling.

Because of this, it is necessary to find out if the stiffeners act assimple supports of the panel (in compression, shear load and combinationload). If this condition is not fulfilled, it must be supposed that thepanel assembly and the stiffeners buckle in a global manner in a mode ofinstability, something which must be avoided in a structure designed foraeronautical use.

A general failure (or global) happens when the structure is no longercapable of supporting additional loads. It can be said therefore thatthe structure has reached failure loading or loading capacity.

General failure covers all types of failure:

-   -   Failure due to an instability (general instability,        post-buckling . . . )    -   Failure caused by exceeding the maximum load supported by the        material (for example after local buckling)

Effective width (or working width) of the skin of a panel is defined asthe portion of the skin which is supported by a longeron in a stiffenedpanel structure which does not buckle when it is subject to an axialcompression load.

Buckling of the skin alone does not constitute a panel failure; thepanel will in fact support additional loads up to the stress at whichthe column formed by the stiffener and the effective panel starts tofail. When the stress in the stiffener goes above the buckling stress ofthe skin, the skin adjacent to the stiffener tolerates an additionalstress because of the support provided by the stiffeners. However, thestress at the centre of the panel will not go above the initial bucklingstress, whatever the stress reached at the level of the stiffener.

The skin is more effective around the position of the stiffeners becausethere is a local support against buckling. At a given level of stress,lower than that of local buckling of the skin, the effective width isequal to the width of the panel. The theory of effective width isillustrated by FIG. 6

Idealisation of Material

It is herewith noted that up to the yield stress (F_(cy)), thestress-strain curve of material is idealised by the known law of Rambergand Osgood (referred to as the R&O formula in the rest of thedescription):

$\begin{matrix}{ɛ = {\frac{\sigma}{E_{c}} + {0.002 \cdot \left( \frac{\sigma}{Fcy} \right)^{n_{c}}}}} & {{Equation}\mspace{14mu} 0\text{-}1}\end{matrix}$

We can deduce the following expressions:

Secant Modulus:

$\begin{matrix}{E_{\sec} = {{\frac{\sigma}{ɛ}\overset{{R\&}O\mspace{14mu} {law}}{}E_{\sec}} = \frac{1}{\frac{1}{E_{c}} + {\frac{0.002}{F_{cy}} \cdot \left( \frac{\sigma}{F_{cy}} \right)^{({n_{c} - 1})}}}}} & {{Equation}\mspace{14mu} 0\text{-}2}\end{matrix}$

Tangent Modulus

$\begin{matrix}\begin{matrix}{\frac{1}{E_{\tan}} = \frac{\partial(ɛ)}{\partial(\sigma)}} \\{= {\frac{n_{c}}{E_{\sec}} + {\frac{1 - n_{c}}{E_{c}}\overset{{R\&}O\mspace{14mu} {law}}{}E_{\tan}}}} \\{= \frac{1}{\frac{1}{E_{c}} + {\frac{0.002}{F_{cy}} \cdot n_{c} \cdot \left( \frac{\sigma}{F_{cy}} \right)^{({n_{c} - 1})}}}}\end{matrix} & {{Equation}\mspace{14mu} 0\text{-}3}\end{matrix}$

Poisson Coefficient:

$\begin{matrix}{v = {{{\frac{E_{\sec}}{E_{c}} \cdot v_{e}} + {{\left( {1 - \frac{E_{\sec}}{E_{c}}} \right) \cdot v_{p}}\mspace{14mu} {With}\mspace{14mu} v_{p}}} = 0.5}} & {{Equation}\mspace{14mu} 0\text{-}4}\end{matrix}$

It is noted that, with the R&O ratio (parameter n or n corrected), knownby those skilled in the art, these equations are only correct in thezone [0; F_(cy)]. For the following part of this study, this zone mustbe extended from F_(cy) to F_(tu). Over F_(cy), different curves can beused up to the ultimate stress, in particular: R&O formula using amodified n coefficient, or elliptical method.

In the following, the R&O formula uses a modified coefficient.Continuity between the two curves is maintained. The modified ncoefficient of the R&O formula is calculated by:

$\begin{matrix}{n = \frac{\ln \left( \frac{ɛ_{2p}}{ɛ_{1p}} \right)}{\ln \left( \frac{\sigma_{2}}{\sigma_{1}} \right)}} & {{Equation}\mspace{11mu} 0\text{-}5}\end{matrix}$

With:

-   -   e_(2p)=ε_(ult)    -   e_(1p)=0.002    -   σ₂=F    -   σ₁=F_(cy)

It is noted that to use this formula, the following criterion must berespected: F_(tu)>F_(cy) and ε_(ult)>0.002

In the elliptical method, above F_(cy), another curve is used up to theultimate stress: the elliptical extension curve. Naturally, thecontinuity between the R&O curve and the elliptical extension curve isensured.

The stress-strain ratios of the elliptical extension are:

$\begin{matrix}{{{{ɛ_{E}(\sigma)} = {ɛ_{3} - {a\sqrt{1 - \frac{\left( {\sigma - \sigma_{E\_ ref} + b} \right)^{2}}{b^{2}}}}}};}{{\sigma_{E}(ɛ)} = {\sigma_{E\_ ref} - b + {b\sqrt{1 - \frac{\left( {ɛ - ɛ_{3}} \right)^{2}}{a^{2}}}}}}} & {{Equation}\mspace{14mu} 0\text{-}6}\end{matrix}$

with:

${a = \sqrt{\frac{{- b^{2}} \cdot x_{1}}{m \cdot \left( {b - D} \right)}}};{b = {\frac{D \cdot \left( {{m \cdot x_{1}} + D} \right)}{{m \cdot x_{1}} + {2\; D}}\mspace{31mu} - {{ellipse}\mspace{14mu} {parameters}}}}$x₁ = ɛ₂ − ɛ₃; D = σ_(E_ref) − σ_(RO_ref);$m = \frac{E}{1 + {ɛ_{RO\_ ref} \cdot \frac{nE}{\sigma_{RO\_ ref}}}}$

Plasticity

Again it is noted that, it is known that plasticity correction factorsdepend on the type of load and boundary conditions.

The plasticity correction factors for flat rectangular panels arepresented in table 1 below.

Boundary Loading conditions Equation Compression and bending Flange witha hinged edge unloaded (free-$\eta_{1} = {\frac{E_{\sec}}{E_{c}} \cdot \frac{1 - v_{e}^{2}}{1 - v^{2}}}$supported edges) Flange with a fixed unloaded edge (clamped-free$\eta_{2} = {\eta_{1} \cdot \left( {0.33 + {0.335 \cdot \sqrt{1 + {3 \cdot \frac{E_{\tan}}{E_{\sec}}}}}} \right)}$edges) Plate with unloaded hinged edges (supported edges)$\eta_{3} = {\eta_{1} \cdot \left( {0.5 + {0.25\sqrt{1 + {3\frac{E_{\tan}}{E_{\sec}}}}}} \right)}$Plate with unloaded fixed edges (clamped edges)$\eta_{4} = {\eta_{1} \cdot \left( {0.352 + {0.324\sqrt{1 + {3\frac{E_{\tan}}{E_{\sec}}}}}} \right)}$Compression Column $\eta_{5} = \frac{E_{\tan}}{E_{c}}$ Shear load allconditions σ_(eq) = τ · {square root over (3)}$\eta_{6} = \frac{G_{\sec}}{G}$ Shear load re-tightened edges σ_(eq) = τ· 2$\eta_{8} = {\eta_{1}\left( {0.83 + {0.17\frac{E_{\tan}}{E_{\sec}}}} \right)}$

In the specific case of shear load, the compression stress-strain curveof the material is equally used as follows:

-   -   Calculation of the equivalent normal stress: σ_(eq)=τ·√{square        root over (3)}    -   Calculation of the corresponding E_(s) and ν values based on        this stress: σ_(eq)

$\eta_{6} = {\frac{G_{s}}{G} = {{\left( \frac{1 + v_{e}}{1 + v} \right) \cdot \frac{E_{\sec}}{E_{c}}} = {\left( \frac{1 + v}{1 + v_{e}} \right) \cdot \eta_{1}}}}$

Step 1—Data Entry Module: Geometry, Material, Loading

The method includes a first phase of entering data relating to the panelstiffened by triangular pockets being considered and to the loadingapplied to this panel. These data are entered using known means andmemorised in a data base which is also of a known type.

The entry parameters for the analytical calculation of panels stiffenedby triangular pockets particularly include:

-   -   General dimensions: rectangular panel (dimensions: L_(x), L_(y))    -   Straight section of stiffeners: dimensions of the web: b, d    -   Constant thickness of the panel (t)    -   Load boundaries of the panel N_(x), N_(y), N_(xy)

Calculation of Mass

This part is designed for the complete calculation of the mass of thepanel stiffened by triangular pockets, including taking into account theradii of the fillet and the node. This step of calculating mass isindependent from the rest of the method described herein. The mass iscalculated in a known manner using the geometrical definition of thepanel.

Data entered for this process are the geometry of the panel includingthe radii of pockets and nodes (R_(n) and R_(f)). Exit data is thepanel's mass.

Mass is calculated by adding up the mass of the skin and the longerons.The radii of filets between two longerons and between the skin and thelongerons are also taken into account. Calculation of mass is based ontwo elementary triangles: an isosceles triangle and a rectangulartriangle (see FIGS. 8, 9 and 10).

Step 2—Calculation of Applied Loads

This step makes it possible to calculate the stresses applied in theskin and the stiffeners based on the geometry of the panel stiffened bytriangular pockets and the external loads. The method takes into accounta plasticity correction of applied loads, done using an iterativeprocess. It makes it possible to take into account the post-buckling ofstiffeners and pockets.

This represents substantial progress in relation to the NASA “Isogrid”design handbook” (NASA-CR-124075, 02/1973) in that it particularly takesinto account the following points: grid of stiffeners with θ≠60, panelstiffened by triangular pockets considered as an assembly of twoorthotropic plates.

The entry data for this step are:

-   -   Geometric data:        -   θ: angle of the base of the triangle,        -   a: base of the triangle,        -   A_(i) ^(st): straight section of the stiffener, i=0°, θ or            −θ.        -   t_(s): thickness of the skin,        -   t_(g): thickness of the panel equivalent to the grid    -   Data on the material:        -   E_(x) ^(s), E_(y) ^(s): Young's modulus of the skin,        -   G_(xy) ^(s): shear modulus of the skin,        -   ν_(xy) ^(s), ν_(yx) ^(s): Poisson coefficient of the skin,        -   E^(st): Young's modulus of the stiffeners,        -   ν^(st): Poisson coefficient of the stiffeners        -   Material data (n: Ramberg & Osgood coefficient, Fcy, Ftu,            ν_(plast)=0.5)    -   Loads applied on the structure (N_(x) ⁰, N_(y) ⁰, N_(xy) ⁰)

The data obtained at the end of this step are:

-   -   N_(x) ^(s), N_(y) ^(s), N_(xy) ^(s): flow in the skin,    -   σ_(x) ^(s), σ_(y) ^(s), τ_(xy) ^(s): stresses in the skin,    -   σ_(0°), σ_(θ), σ_(−θ): stresses in the stiffeners,    -   F_(0°), F_(θ), F_(−θ): loads in the stiffeners.

In the following part of the description, the skin is assumed to be ofan isotropic material.

The method provides entries for:

-   -   The analysis of resistance (step 4): stresses in the skin and in        the stiffeners    -   the analysis of pocket buckling (step 5.1): stresses in the skin    -   the analysis of stiffener buckling (step 5.2): stresses in the        stiffeners    -   the analysis of general instability (step 6): stresses in the        skin and in the stiffeners to calculate the bending rigidity of        the panel stiffened by triangular pockets.

The calculation method requires entry data on the post-buckling ofstiffeners: A_(0°) ^(st), A_(+θ) ^(st) and A_(−θ) ^(st) and onpost-buckling of pockets: t_(s) _(_) _(eff)

The method takes into account the redistribution of applied stressesbetween the panel and the grid of stiffeners due in the first instance,to the post-buckling of the stiffeners, by the definition of aneffective straight section for each type of stiffener (0°, +θ or −θ):A_(0°) ^(st), A_(+θ) ^(st) and A_(−θ) ^(st), and in the second instance,to the post-buckling of the pocket through an effective thickness of thepanel: t_(s) _(_) _(eff), finally, to the plasticity of applied externalloads, using an iterative process on the different properties of thematerial: E_(0°) ^(st), E_(+θ) ^(st), E_(−θ) ^(st) for the stiffenersand E_(x) ^(s), E_(y) ^(s) and ν_(ep) ^(st) for the skin.

The external load is assumed to be in the plane of the panel and appliedat the centre of gravity of the section:

$\mspace{20mu} {\begin{Bmatrix}N \\M\end{Bmatrix} = {\begin{bmatrix}A & B \\B & C\end{bmatrix} \cdot \begin{Bmatrix}ɛ \\\kappa\end{Bmatrix}}}$${{{therefore}\text{:}\mspace{14mu} ɛ} \neq {0\mspace{14mu} {and}\mspace{14mu} \kappa}} = {{0\mspace{11mu} \mspace{14mu} \left\{ N \right\}} = {{{\lbrack A\rbrack \cdot \left\{ ɛ \right\}}\mspace{14mu} {with}\mspace{14mu} \left\{ N \right\}} = \begin{Bmatrix}{Nx} \\{Ny} \\{Nxy}\end{Bmatrix}}}$

In consequence, the stresses in the skin do not depend on the thicknessof said skin and the position in the plane. Also, stresses in thestiffeners do not depend on the position on the section of thestiffener, but only on the angle of the stiffener.

The geometric definition of the grid of stiffeners used for carrying outcalculations is defined in FIG. 11:

To obtain a panel stiffened by triangular pockets, this elementary shapeis associated with the skin and is repeated as many times as isrequired. Because of this, this method does not take into account theconcept of the geometry of the edges.

For each stiffener, the real section (A_(i) ^(st) with i: 0°, +θ or −θ)is given by the ratio: A_(i) ^(st)=% A_(i)×A^(st) (in the presentnon-limitative example, only the case of % A_(i)=1 is envisaged).

The straight section of the stiffeners includes the section of theradius of the pocket

$\left( {2 \cdot {R_{f}^{2}\left( {1 - \frac{\pi}{4}} \right)}} \right).$

Whatever their position on the grid, the stresses and strains areidentical for each type of stiffener (0°, +θ, −θ).

To take into account the plasticity which can occur in each stiffener,the Young's modulus is specific to each type of stiffener (0°, +θ, −θ):E_(0°) ^(st), E_(+θ) ^(st), E_(−θ) ^(st).

The “material” matrix E is defined by:

$\overset{\_}{\overset{\_}{E_{g}}} = {\begin{pmatrix}E_{0{^\circ}}^{st} & 0 & 0 \\0 & E_{\theta}^{st} & 0 \\0 & 0 & E_{- \theta}^{st}\end{pmatrix}.}$

Step 3—Calculation of Internal Loads

3.1 Plate Equivalent to the Stiffeners

3.1.1 Relation Between Global Strains and Stiffener Strains

The geometric notations and conventions are illustrated by FIG. 16. Weare looking for the relation between (ε_(x), ε_(y), E_(xy)) and (ε_(0°),ε_(θ), ε_(−θ)). General strains are defined by the following formulas:

ε_(nn) ={right arrow over (n)}·ε·{right arrow over (n)}

Therefore our strains are:

$ɛ_{0{^\circ}} = {{\overset{\rightarrow}{i_{0{^\circ}}} \cdot \overset{\_}{\overset{\_}{ɛ}} \cdot \overset{\rightarrow}{i_{0{^\circ}}}} = {\begin{pmatrix}1 & 0 & 0\end{pmatrix} \cdot \begin{pmatrix}ɛ_{xx} & ɛ_{xy} & ɛ_{xz} \\ɛ_{yx} & ɛ_{yy} & ɛ_{yz} \\ɛ_{zx} & ɛ_{zy} & ɛ_{zz}\end{pmatrix} \cdot \begin{pmatrix}1 \\0 \\0\end{pmatrix}}}$$ɛ_{\theta} = {{\overset{\rightarrow}{i_{\theta}} \cdot \overset{\_}{\overset{\_}{ɛ}} \cdot \overset{\rightarrow}{i_{\theta}}} = {\begin{pmatrix}{\cos \; \theta} & {\sin \; \theta} & 0\end{pmatrix} \cdot \begin{pmatrix}ɛ_{xx} & ɛ_{xy} & ɛ_{xz} \\ɛ_{yx} & ɛ_{yy} & ɛ_{yz} \\ɛ_{zx} & ɛ_{zy} & ɛ_{zz}\end{pmatrix} \cdot \begin{pmatrix}{\cos \; \theta} \\{\sin \; \theta} \\0\end{pmatrix}}}$$ɛ_{- \theta} = {{\overset{\rightarrow}{i_{- \theta}} \cdot \overset{\_}{\overset{\_}{ɛ}} \cdot \overset{\rightarrow}{i_{- \theta}}} = {\begin{pmatrix}{\cos \; \theta} & {{- \sin}\; \theta} & 0\end{pmatrix} \cdot \begin{pmatrix}ɛ_{xx} & ɛ_{xy} & ɛ_{xz} \\ɛ_{yx} & ɛ_{yy} & ɛ_{yz} \\ɛ_{zx} & ɛ_{zy} & ɛ_{zz}\end{pmatrix} \cdot \begin{pmatrix}{\cos \; \theta} \\{{- \sin}\; \theta} \\0\end{pmatrix}}}$

And finally:

$\begin{matrix}{\begin{pmatrix}ɛ_{0{^\circ}} \\ɛ_{\theta} \\ɛ_{- \theta}\end{pmatrix} = {\begin{pmatrix}1 & 0 & 0 \\{\cos^{2}\theta} & {\sin^{2}\theta} & {2\; \sin \; \theta \; \cos \; \theta} \\{2\; \cos^{2}\theta} & {2\; \sin^{2}\theta} & {{- 2}\; \sin \; \theta \; \cos \; \theta}\end{pmatrix} \cdot \begin{pmatrix}ɛ_{x} \\ɛ_{y} \\ɛ_{xy}\end{pmatrix}}} & {{Equation}\mspace{14mu} 3\text{-}1}\end{matrix}$

The above matrix is denoted Z:

$\overset{\_}{\overset{\_}{Z}} = \begin{pmatrix}1 & 0 & 0 \\{\cos^{2}\theta} & {\sin^{2}\theta} & {2\; \sin \; \theta \; \cos \; \theta} \\{2\; \cos^{2}\theta} & {2\; \sin^{2}\theta} & {{- 2}\; \sin \; \theta \; \cos \; \theta}\end{pmatrix}$

3.1.2 Relation Between Stresses and Strains

As stated, the geometric notations and conventions are illustrated byFIG. 16. The loads in stiffeners are given by the following expressions:

{right arrow over (P)} _(i)=ε_(i) ·E ^(st) ·A ^(st) _(i) ·{right arrowover (i)} _(i),(i=0°,θ,−θ)   Equation 3-2

Therefore the base element below is subjected to: {right arrow over(P_(0°))}, {right arrow over (P_(θ))}, {right arrow over(P_(−θ))}({right arrow over (P_(0°))} counted twice because thedimension of the base element according axis Y is 2 h, therefore thestiffener corresponding to 0° should also be taken into account).

According to Axis x:

$P_{x}^{g} = {{\sum\limits_{i}\; {{\overset{\rightarrow}{P}}_{i} \cdot \overset{\rightarrow}{x}}} = {{\left( {{2\overset{\rightarrow}{P_{0{^\circ}}}} + \overset{\rightarrow}{P_{\theta}} + \overset{\rightarrow}{P_{- \theta}}} \right) \cdot \overset{\rightarrow}{x}} = {{2\; E_{0{^\circ}}^{st}A_{0{^\circ}}^{st}ɛ_{0{^\circ}}} + {E_{\theta}^{st}A_{\theta}^{st}ɛ_{\theta}\cos \; \theta} + {E_{\_\theta}^{st}A_{- \theta}^{st}ɛ_{- \theta}\cos \; \theta}}}}$

According to Axis y:

P _(y) ^(g) =E _(θ) ^(st) A _(θ) ^(st)ε_(θ) sin θ+E _(−θ) ^(st) A _(−θ)^(st)ε_(−θ) sin θ

Shear Load in the Plane ({right arrow over (x)},{right arrow over (y)}):

P _(xy) ^(g) =E _(θ) ^(st) A _(θ) ^(st)ε_(θ) sin θ−E _(−θ) ^(st) A _(−θ)^(st)ε_(−θ) sin θ   Equation 3-3

To obtain the stresses, the load is divided by the surface of a baseelement. The section of the base element on a normal surface accordingto axis X is 2ht_(g)=a tan θ·t_(g) The section of the base element on anormal surface according to axis Y is at_(g).

In terms of stresses we have:

$\sigma_{x}^{g} = {\frac{P_{x}^{g}}{2\; {ht}_{g}} = {\frac{1}{2\; {ht}_{g}}{\begin{pmatrix}{2\; E_{0{^\circ}}^{st}A_{0{^\circ}}^{st}} & {E_{\theta}^{st}A_{\theta}^{st}\cos \; \theta} & {E_{- \theta}^{st}A_{- \theta}^{st}\cos \; \theta}\end{pmatrix} \cdot \begin{pmatrix}ɛ_{0{^\circ}} \\ɛ_{\theta} \\ɛ_{- \theta}\end{pmatrix}}}}$ $\sigma_{x}^{g} = {\frac{1}{{at}_{g}}{\begin{pmatrix}\frac{2\; E_{0{^\circ}}^{st}A_{0{^\circ}}^{st}}{\tan \; \theta} & {E_{\theta}^{st}A_{\theta}^{st}\frac{\cos^{2}\; \theta}{\sin \; \theta}} & {E_{- \theta}^{st}A_{- \theta}^{st}\frac{\cos^{2}\; \theta}{\sin \; \theta}}\end{pmatrix} \cdot \begin{pmatrix}ɛ_{0{^\circ}} \\ɛ_{\theta} \\ɛ_{- \theta}\end{pmatrix}}}$

For σ_(y) and σ_(xy), we obtain with the same method:

$\sigma_{y}^{g} = {\frac{1}{{at}_{g}}{\begin{pmatrix}0 & {E_{\theta}^{st}A_{\theta}^{st}\sin \; \theta} & {E_{- \theta}^{st}A_{- \theta}^{st}\sin \; \theta}\end{pmatrix} \cdot \begin{pmatrix}ɛ_{0{^\circ}} \\ɛ_{\theta} \\ɛ_{- \theta}\end{pmatrix}}}$ $\tau_{xy}^{g} = {\frac{1}{{at}_{g}}{\begin{pmatrix}0 & {E_{\theta}^{st}A_{\theta}^{st}\sin \; \theta} & {{- E_{- \theta}^{st}}A_{- \theta}^{st}\sin \; \theta}\end{pmatrix} \cdot \begin{pmatrix}ɛ_{0{^\circ}} \\ɛ_{\theta} \\ɛ_{- \theta}\end{pmatrix}}}$

The same results can be presented in matrix form:

$\begin{matrix}{\begin{pmatrix}\sigma_{x}^{g} \\\sigma_{y}^{g} \\\tau_{xy}^{g}\end{pmatrix} = {\begin{pmatrix}\frac{2\; E_{0{^\circ}}^{st}A_{0{^\circ}}^{st}}{\tan \; \theta} & {E_{\theta}^{st}A_{\theta}^{st}\frac{\cos^{2}\; \theta}{\sin \; \theta}} & {E_{- \theta}^{st}A_{- \theta}^{st}\frac{\cos^{2}\; \theta}{\sin \; \theta}} \\0 & {E_{\theta}^{st}A_{\theta}^{st}\sin \; \theta} & {E_{- \theta}^{st}A_{\theta}^{st}\sin \; \theta} \\0 & {E_{\theta}^{st}A_{\theta}^{st}\sin \; \theta} & {{- E_{- \theta}^{st}}A_{\theta}^{st}\sin \; \theta}\end{pmatrix} \cdot \begin{pmatrix}ɛ_{0{^\circ}} \\ɛ_{\theta} \\ɛ_{- \theta}\end{pmatrix}}} & {{Equation}\mspace{14mu} 3\text{-}4}\end{matrix}$

The above matrix is denoted T:

$\overset{\_}{\overset{\_}{T}} = {\frac{1}{{at}_{g}}\begin{pmatrix}\frac{2\; E_{0{^\circ}}^{st}A_{0{^\circ}}^{st}}{\tan \; \theta} & {E_{\theta}^{st}A_{\theta}^{st}\frac{\cos^{2}\; \theta}{\sin \; \theta}} & {E_{- \theta}^{st}A_{- \theta}^{st}\frac{\cos^{2}\; \theta}{\sin \; \theta}} \\0 & {E_{\theta}^{st}A_{\theta}^{st}\sin \; \theta} & {E_{- \theta}^{st}A_{\theta}^{st}\sin \; \theta} \\0 & {E_{\theta}^{st}A_{\theta}^{st}\sin \; \theta} & {{- E_{- \theta}^{st}}A_{\theta}^{st}\sin \; \theta}\end{pmatrix}}$

Thus by using Equation 3-1 and the Z matrix notation:

$\begin{matrix}{\begin{pmatrix}\sigma_{x}^{g} \\\sigma_{y}^{g} \\\tau_{xy}^{g}\end{pmatrix} = {\overset{\_}{\overset{\_}{T}} \cdot \overset{\_}{\overset{\_}{Z}} \cdot \begin{pmatrix}ɛ_{x} \\ɛ_{y} \\ɛ_{xy}\end{pmatrix}}} & {{Equation}\mspace{14mu} 3\text{-}5}\end{matrix}$

The previous matrix is denoted W: W=T·Z

This relation (Equation 3-5) signifies that the behaviour of the panelequivalent to the stiffeners is similar to an anisotropic material (theW matrix can be completed: all these cells have non-null values).

3.2 Stiffened Panels

We use the Kirchhoff hypothesis: plane sections remain plane afterstraining. The network of stiffeners is modelled by an equivalent panelwith a W matrix behaviour (see Equation 3-5). This disposition ofmodelling a panel stiffened by triangular pockets, by two orthotropicplates is illustrated by FIG. 15.

For the calculation of ν_(yx), we have:

$\frac{v_{xy}^{s}}{E_{x}^{s}} = \frac{v_{yx}^{s}}{E_{y}^{s}}$

3.2.1. Stresses and Loads of Stiffeners

Flow in the Panel Equivalent to the Stiffeners

The general expression of flows is:

$\begin{matrix}{N_{\alpha \; \beta} = {\int_{h}\ {\sigma_{\; {\alpha\beta}}{dz}}}} & {{Equation}\mspace{14mu} 3\text{-}6}\end{matrix}$

The flow according to axis X is expressed as follows:

$N_{xx} = {{\int_{h}\ {\sigma_{xx}{dz}}} = {{\int_{- \frac{h}{2}}^{{- \frac{h}{2}} + t_{s}}\ {\sigma_{xx}^{s}{dz}}} + {\int_{{- \frac{h}{2}} + t_{s}}^{\frac{h}{2}}{\sigma_{xx}^{g}{dz}}}}}$$N_{xx} = {{\frac{E_{s}^{x}t_{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}} \cdot ɛ_{xx}} + {\frac{v_{xy}^{s}E_{y}^{s}t_{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}} \cdot ɛ_{yy}} + {\sigma_{xx}^{g} \cdot t_{g}}}$

by using Equation 3-5:

$N_{x} = {{\left\lbrack {\frac{E_{x}^{s}t_{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}} + {t_{g} \cdot {\overset{\_}{\overset{\_}{W}}}_{1,1}}} \right\rbrack \cdot ɛ_{x}} + {\left\lbrack {\frac{V_{xy}^{s}E_{y}^{s}t_{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}} + {t_{g} \cdot {\overset{\_}{\overset{\_}{W}}}_{1,2}}} \right\rbrack \cdot ɛ_{y}} + {t_{g} \cdot {\overset{\_}{\overset{\_}{W}}}_{1,3} \cdot ɛ_{xy}}}$

And, by using the same method for the N_(y) and N_(xy) flows.

$\begin{matrix}{{N_{y} = {{\left\lbrack {\frac{v_{yx}^{s}E_{x}^{s}t_{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}} + {t_{g} \cdot {\overset{\_}{\overset{\_}{W}}}_{2,1}}} \right\rbrack \cdot ɛ_{x}} + {\left\lbrack {\frac{E_{y}^{s}t_{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}} + {t_{g} \cdot {\overset{\_}{\overset{\_}{W}}}_{2,2}}} \right\rbrack \cdot ɛ_{y}} + {t_{g} \cdot {\overset{\_}{\overset{\_}{W}}}_{2,3} \cdot ɛ_{xy}}}}{N_{xy} = {{t_{g} \cdot {\overset{\_}{\overset{\_}{W}}}_{3,2} \cdot ɛ_{x}} + {t_{g} \cdot {\overset{\_}{\overset{\_}{W}}}_{3,3} \cdot ɛ_{y}} + {\left( {{2\; G_{xy}t_{s}} + {t_{g} \cdot {\overset{\_}{\overset{\_}{W}}}_{3,1}}} \right) \cdot ɛ_{xy}}}}} & {{Equation}\mspace{14mu} 3\text{-}7}\end{matrix}$

These expressions clearly show the distribution of flow between the skinand the panel equivalent to the stiffeners. In the skin, the relationbetween the flows and strains is:

$\begin{matrix}{\begin{pmatrix}N_{x}^{s} \\N_{y}^{s} \\N_{xy}^{s}\end{pmatrix} = {{\begin{pmatrix}N_{x} \\N_{y} \\N_{xy}\end{pmatrix} - \begin{pmatrix}N_{x}^{g} \\N_{y}^{g} \\N_{xy}^{g}\end{pmatrix}} = {\begin{pmatrix}\frac{E_{x}^{s}t_{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}} & \frac{v_{yx}^{s}E_{y}^{s}t_{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}} & 0 \\\frac{v_{yx}^{s}E_{x}^{s}t_{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}} & \frac{E_{y}^{s}t_{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}} & 0 \\0 & 0 & {2\; G_{xy}^{s}t_{s}}\end{pmatrix} \cdot \begin{pmatrix}ɛ_{x} \\ɛ_{y} \\ɛ_{xy}\end{pmatrix}}}} & {{Equation}\mspace{14mu} 3\text{-}8}\end{matrix}$

We note X the previous matrix:

$\begin{matrix}{{\overset{\_}{\overset{\_}{X}} = \begin{pmatrix}\frac{E_{x}^{s}t_{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}} & \frac{v_{yx}^{s}E_{y}^{s}t_{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}} & 0 \\\frac{v_{yx}^{s}E_{x}^{s}t_{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}} & \frac{E_{y}^{s}t_{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}} & 0 \\0 & 0 & {2\; G_{xy}^{s}t_{s}}\end{pmatrix}}{{Thus}\text{:}}{\begin{pmatrix}N_{x} \\N_{y} \\N_{xy}\end{pmatrix} = {\begin{pmatrix}N_{x}^{g} \\N_{y}^{g} \\N_{xy}^{g}\end{pmatrix} + {\frac{1}{t_{g}} \cdot \overset{\_}{\overset{\_}{X}} \cdot {\overset{\_}{\overset{\_}{W}}}^{- 1} \cdot \begin{pmatrix}N_{x}^{g} \\N_{y}^{g} \\N_{xy}^{g}\end{pmatrix}}}}} & {{Equation}\mspace{14mu} 3\text{-}9}\end{matrix}$

By inversing this relation, the flows in the grid are expressedaccording to the globally applied flows:

$\begin{matrix}{{\begin{pmatrix}N_{x}^{g} \\N_{y}^{g} \\N_{xy}^{g}\end{pmatrix} = {V \cdot \begin{pmatrix}N_{x} \\N_{y} \\N_{xy}\end{pmatrix}}}{{With}\text{:}}{\overset{\_}{\overset{\_}{V}} = \left( {\overset{\_}{\overset{\_}{I_{d}}} + {\frac{1}{t_{g}} \cdot \overset{\_}{\overset{\_}{X}} \cdot {\overset{\_}{\overset{\_}{W}}}^{- 1}}} \right)^{- 1}}\left( {\overset{\_}{\overset{\_}{I_{d}}}\mspace{20mu} {is}\mspace{14mu} {the}\mspace{14mu} {identity}\mspace{14mu} {matrix}} \right)} & {{Equation}\mspace{14mu} 3\text{-}10}\end{matrix}$

Stresses and Loads in Stiffeners

The flow in the panel equivalent to the stiffeners can be expressed by:

$\begin{matrix}{\begin{pmatrix}N_{x}^{g} \\N_{y}^{g} \\N_{xy}^{g}\end{pmatrix} = {t_{g} \cdot \overset{\_}{\overset{\_}{T}} \cdot \begin{pmatrix}ɛ_{0{^\circ}} \\ɛ_{\theta} \\ɛ_{- \theta}\end{pmatrix}}} & {{Equation}\mspace{14mu} 3\text{-}11} \\{\begin{pmatrix}N_{x}^{g} \\N_{y}^{g} \\N_{xy}^{g}\end{pmatrix} = {t_{g} \cdot \overset{\_}{\overset{\_}{T}} \cdot {\overset{\_}{\overset{\_}{E_{g}}}}^{- 1} \cdot \begin{pmatrix}\sigma_{0{^\circ}} \\\sigma_{\theta} \\\sigma_{- \theta}\end{pmatrix}}} & {{Equation}\mspace{14mu} 3\text{-}12}\end{matrix}$

By using the following notation: U=t_(g)·T·E_(g) ⁻¹ We have:

$\begin{matrix}{\begin{pmatrix}\sigma_{0{^\circ}} \\\sigma_{\theta} \\\sigma_{- \theta}\end{pmatrix} = {U^{- 1} \cdot \begin{pmatrix}N_{x}^{g} \\N_{y}^{g} \\N_{xy}^{g}\end{pmatrix}}} & {{Equation}\mspace{14mu} 3\text{-}13}\end{matrix}$

Finally, the loads and stresses in the stiffeners are expressedaccording to the flows of external loads:

$\begin{matrix}{\begin{pmatrix}\sigma_{0{^\circ}} \\\sigma_{\theta} \\\sigma_{- \theta}\end{pmatrix} = {{{U^{- 1} \cdot V \cdot \begin{pmatrix}N_{x} \\N_{y} \\N_{xy}\end{pmatrix}}\mspace{14mu} {{and}\begin{pmatrix}F_{0{^\circ}} \\F_{\theta} \\F_{- \theta}\end{pmatrix}}} = {\begin{pmatrix}A_{0{^\circ}}^{st} \\A_{\theta}^{st} \\A_{- \theta}^{st}\end{pmatrix}\begin{pmatrix}\sigma_{0{^\circ}} \\\sigma_{\theta} \\\sigma_{- \theta}\end{pmatrix}}}} & {{Equation}\mspace{14mu} 3\text{-}14}\end{matrix}$

3.2.2 Flows and Stresses in the Skin

As shown by Equation 3-8, flows in the skin are expressed as below:

$\begin{matrix}{\begin{pmatrix}N_{x}^{s} \\N_{y}^{s} \\N_{xy}^{s}\end{pmatrix} = {\begin{pmatrix}N_{x} \\N_{y} \\N_{xy}\end{pmatrix} - \begin{pmatrix}N_{x}^{g} \\N_{y}^{g} \\N_{xy}^{g}\end{pmatrix}}} & {{Equation}\mspace{14mu} 3\text{-}15}\end{matrix}$

thus, the stresses in the skin are expressed by:

$\begin{matrix}{\begin{pmatrix}\sigma_{x}^{s} \\\sigma_{y}^{s} \\\tau_{xy}^{s}\end{pmatrix} = {\frac{1}{t_{s}} \cdot \begin{pmatrix}N_{x}^{s} \\N_{y}^{s} \\N_{xy}^{s}\end{pmatrix}}} & {{Equation}\mspace{14mu} 3\text{-}16}\end{matrix}$

3.3 The method of calculating plasticity corrected applied loads ispresented here with references to the matrices introduced in thedescription (FIG. 17).

We note that the theory used for calculating plasticity supposes theisotropic quality of the skin which forms the skin. The solution forplasticity corrected applied loads is provided by an iterative method.

A process of convergence must be carried out until the five materialparameters (E_(0°) ^(st), E_(+θ) ^(st), E_(−θ) ^(st), E_(skin), ν_(ep))entered at the start of the iterative process are equal to the sameparameters calculated at the exit (after calculation of plastic stress).In FIG. 17, already cited, the convergence parameters are indicated by agrey background.

More precisely, in this iterative process, the initial entries are theloads applied in the grid of stiffeners and in the skin.

For the grid of stiffeners, the data of the n^(th) iteration of Young'smodulus of stiffeners E_(0°) ^(st), E_(+θ) ^(st), E_(−θ) ^(st) in 3directions: 0°, +θ, −θ, make it possible, in association with the valueof the angle θ and the geometry, to calculate the matrix [T] (equation3-4). The values of the angle θ and geometry supplying the matrix [Z](equation 3-1). The matrices [T] and [Z] giving the matrix [W](equation3-5).

For the skin, the material data of the isotropic skin E_(skin), ν_(ep))make it possible to calculate the matrix [X] (equation 3-8).

The [W] and [X] matrices make it possible to calculate the [U](equation3-13) and [V] matrices (equation 3-10).

The results drawn from these matrices include: the flows, elasticstresses, plasticity corrections on stresses, the values of the n^(th)+1iteration of the Young's modulus of corrected stiffeners and skin, andof the Poisson coefficient of the corrected skin, and the loads in thestiffeners.

We understand that the calculation is iterated until the Young's modulusand Poisson coefficients vary during an iteration of a value which islower than a predetermined threshold.

Impact of Plasticity Correction on the Calculation of General Buckling:

Naturally, plasticity correction changes the behaviour law matrixcalculated in the general instability modulus throughout the 5 materialparameters (see the section on general instability).

Because of this, plasticity correction also alters the coefficientsΩ_(i) (i=1 . . . 3) used for calculating general buckling. Plasticitycorrection in the analysis of general buckling is provided by thesemodified coefficients Ω_(i).

3.4 Example: Distribution of Loads Under Bi-Compression and Shear Load

In the step of calculating the applied stresses the radius of the pocketfillet is taken into account for the calculation of the stiffenersection. In addition, there is no post-buckling, we can therefore write:

% A _(0°) ^(st)=% A _(+e) ^(st)=% A _(−e) ^(st)=100%

t _(s) _(_) _(eff) =t _(s)

In the present non-limitative example described herein, the geometry ofthe panel stiffened by triangular pockets is defined by:

L_(x) = 1400.45 mm a = 198 mm node radius: R_(n) = 9 mm L_(y) = 685.8 mmt = 3.64 mm pocket radius: R_(f) = 4 mm θ = 58° b = 2.5 mm d = 37.36 mm

We consider an isotropic material. The elasto-plastic law used is thatof Ramberg & Osgood.

E 78000 Nu 0.3 Ftu 490 Fty 460 e % 20% n 40

N_(x)=−524.65 N/mm

N_(y)=−253.87 N/mm

N_(z)=327.44 N/mm

The method of calculating internal loads and applied loads includingplasticity correction is written in the form of a matrix:

The obtained results are therefore the flows, stresses and loads in thestiffeners:

N_(xg) = −81.1 N/mm σ_(0°) = −101.14 MPa F_(0°) = −104417N N_(yg) = −39N/mm σ_(+θ) = 44.25 MPa F_(+θ) = 4437N N_(xyg) = 48.1 N/mm σ_(−θ) =−135.07 MPa F_(−θ) = −13537N

As well as the flows and stresses in the skin:

N_(xs)=−443.5 N/mm σ_(xs)=−121.85 MPa

N_(ys)=−214.87 N/mm σ_(ys)=−59.03 MPa

N_(xys)=279.3 N/mm τ_(xys)=76.74 MPa

In this example, the applied stresses reside in the elastic domain.

The flow distribution between the skin and the grid of stiffeners issummarised by the following table:

Distribution of loads - Distribution of loads - Flow External flowspercentages (N/mm) flows Grid Skin Grid Skin Nx −524.65 −81.13 −443.5215.46% 84.54% Ny −253.87 −39.00 −214.87 15.36% 84.64% Nxy +327.44 +48.12+279.32  14.7%  85.3%

Step 4—Resistance Analysis Module:

This phase is aimed at calculating the reserve factors (RF) bycomparison with the applied loads calculated in the components of thepanel stiffened by triangular pockets, and the maximum stress capacityof the material.

The stress capacities in the ultimately loaded material are determinedby: F_(tu) (the material's ultimate tension resistance) compared to thestresses applied in the stiffener webs, F_(tu) compared to the principalstresses applied in the skin, F_(su) (the material's ultimate shearresistance) compared to the maximum shear capacity in the skin.

Analysis of resistance consists of calculating the reserve factors ofthe limit loaded and ultimate loaded material. The applied stresses comefrom loads in the plane (compression, shear load) or outside of theplane (pressure).

The entry data for this calculation are:

-   -   Capacity values for the material: F_(ty), F_(cy), F_(sy),        F_(tu), F_(su)    -   Stresses applied to the structure:        -   Stresses on the skin (σ_(xs), σ_(ys) et τ_(xys))        -   Normal stresses in the stiffeners

Note: applied stresses are corrected for plasticity in the appliedstress calculation method, as was stated above.

The exit data are the reserve factors.

The analysis of resistance in the plane of the panel is based on thefollowing hypotheses. The stresses in the skin do not depend on thethickness of the skin and the position in the plane. The stresses in astiffener do not depend on the position on the section of the stiffener,but only on the angle of the stiffener.

These hypotheses are not valid when post-buckling and behaviour outsideof the plane are equally taken into account. In these cases, the max/minfunctions, of known types, must be implemented to take into accountthese phenomena.

Calculation of Principal Stresses:

To calculate the skin's reserve factor, the principal stresses (σ_(max),σ_(min) and τ_(max)) are used:

$\begin{matrix}{{{\sigma_{\max \_ s} = {\frac{\sigma_{xs} + \sigma_{ys}}{2} + \sqrt{\frac{\left( {\sigma_{xs} - \sigma_{ys}} \right)^{2}}{2} + \tau_{xys}^{2}}}}\sigma_{\min \_ s} = {\frac{\sigma_{xs} + \sigma_{ys}}{2} - \sqrt{\frac{\left( {\sigma_{xs} - \sigma_{ys}} \right)^{2}}{2} + \tau_{xys}^{2}}}}{\tau_{\max} = \frac{\sigma_{\max \_ s} - \sigma_{\min \_ s}}{2}}} & {{Equation}\mspace{14mu} 4\text{-}1}\end{matrix}$

The value σ_(max) used in the calculation of the reserve factor isdefined as the absolute maximum between σ_(max) _(_) _(s) and σ_(min)_(_) _(s) calculated in Error! Reference source not found.1.

Reserve Factor at Load Limit (LL):

Reserve Factor on the Stiffener Webs:

${RF}_{{{in}\_ {plan}e}{\_ {LL}}}^{{{blade}\_}0} = \frac{F_{y}}{\sigma_{{{blade}\_}0{\_ {LL}}}}$${RF}_{{{in}\_ {plan}e}{\_ {LL}}}^{{{blade}\_} + \theta} = \frac{F_{y}}{\sigma_{{{blade}\_} + {{\theta\_}{LL}}}}$${RF}_{{{in}\_ {plan}e}{\_ {LL}}}^{{{blade}\_} - \theta} = \frac{F_{y}}{\sigma_{{{blade}\_} - {{\theta\_}{LL}}}}$

Reserve Factor on the Skin:

Shear capacity:

${RF}_{{{in}\_ {plan}e}{\_ {LL}}}^{{skin}\_ {shear}} = \frac{F_{sy}}{\tau_{\max \_ {LL}}}$

In this formula, if F_(sy) is unknown, F_(su)/√3 can be used

Principal Stress:

${RF}_{{{in}\_ {plan}e}{\_ {LL}}}^{skin} = \frac{F_{y}}{\sigma_{\max \_ {LL}}}$

-   -   An envelope reserve factor is calculated at load limit:

$\begin{matrix}{{RF}_{{material}\_ {LL}} = {\min \left\{ {{RF}_{{{in}\_ {plan}e}{\_ {LL}}}^{{{blade}\_}0};{RF}_{{{in}\_ {plan}e}{\_ {LL}}}^{{{blade}\_} + \theta};{RF}_{{{in}\_ {plan}e}{\_ {LL}}}^{{{blade}\_} - \theta};{RF}_{{{in}\_ {plan}e}{\_ {LL}}}^{{skin}\_ {shear}};{RF}_{{{in}\_ {plan}e}{\_ {LL}}}^{skin}} \right\}}} & {{Equation}\mspace{14mu} 4\text{-}2}\end{matrix}$

Reserve Factor at Ultimate Load UL:

Reserve Factor on the Stiffener Webs:

${RF}_{{{in}\_ {plan}e}{\_ {UL}}}^{{{blade}\_}0} = \frac{F_{u}}{\sigma_{{{blade}\_}0{\_ {UL}}}}{{(*}{{){RF}_{{{in}\_ {plane}}{\_ {UL}}}^{{{blade}\_} + \theta}} = \frac{F_{u}}{\sigma_{{{blade}\_} + {{\theta\_}{UL}}}}{{(*}{{){RF}_{{{in}\_ {plane}}{\_ {UL}}}^{{{blade}\_} - \theta}} = \frac{F_{u}}{\sigma_{{{blade}\_} - {{\theta\_}{UL}}}}}}}}$

Note: if F_(cu) is unknown, F_(cy) or F_(tu) can be used

Reserve Factor on the Skin:

Maximum Shear Capacity:

${RF}_{{{in}\_ {plan}e}{\_ {UL}}}^{{skin}\_ {shear}} = \frac{F_{su}}{\tau_{\max \_ {UL}}}$

Principal Stress:

${RF}_{{{in}\_ {plan}e}{\_ {UL}}}^{skin} = \frac{F_{u}}{\sigma_{\max \_ {UL}}}$

-   -   An envelope reserve factor is calculated at ultimate load:

$\begin{matrix}{{RF}_{{material}\_ {UL}} = {\min \left\{ {{RF}_{{{in}\_ {plan}e}{\_ {UL}}}^{{{blade}\_}0};{RF}_{{{in}\_ {plan}e}{\_ {UL}}}^{{{blade}\_} + \theta};{RF}_{{{in}\_ {plan}e}{\_ {UL}}}^{{{blade}\_} - \theta};{RF}_{{{in}\_ {plan}e}{\_ {UL}}}^{{skin}\_ {shear}};{RF}_{{{in}\_ {plan}e}{\_ {UL}}}^{skin}} \right\}}} & \;\end{matrix}$

The analysis of resistance outside of the plane of the panel liesoutside of the scope of the present invention.

In an example of an implementation of this part of the method foranalysing resistance in the plane, the same geometry and the same caseof loads as in the preceding sections is studied. In the first instance,the ratio of the loads factor is assumed to be equal to 1.

Reserve factor of the stiffener webs:

-   -   RF_(in) _(_) _(plane) _(_) _(UL) ^(blade) ^(_) ⁰=4.71    -   RF_(in) _(_) _(plane) _(_) _(UL) ^(blade) ^(_) ^(+θ)=11.14    -   RF_(in) _(_) _(plane) _(_) _(UL) ^(blade) ^(_) ^(−θ)=3.63

Reserve Factor of the Skin:

Calculation of Principal Stresses:

-   -   σ_(max) _(_) _(s)=−7.52 MPa    -   σ_(min) _(_) _(s)=−173.35 MPa    -   τ_(max)=82.91 MPa    -   σ_(max) _(_) _(skin) _(_) _(UL)=173.35 MPa

Shear capacity: RF_(in) _(_) _(plane) _(_) _(UL) ^(skin) ^(_)^(shear)=3.41

Principal stress: RF_(in) _(_) _(plane) _(_) _(UL) ^(skin)=2.82

Reserve factor envelope RF at ultimate load (UL)

-   -   RF_(materal) _(_) _(UL)=2.82

In the case of plastic calculation, an iterative calculation is carriedout until the applied loads reach the resistance failure load (in orderto carry out plasticity correction on these). At each iteration loop,the same calculations as previously described are carried out.

Step 5—Calculating Local Stress Capacity,

Two types of instability weaken the global rigidity of the structurestiffened by triangular pockets, but do not cause the global failure ofthe complete structure

-   -   Instability of the panel: buckling of triangular pockets    -   Instability of stiffeners: buckling of stiffener webs

The buckled sections can only support a part of the load which theycould support before they were buckled. Because of this, the appliedloads are redistributed in the structure.

It is noted that in the present invention, post-buckling calculationsare not processed. Because of this, the two types of buckling referredto above are considered as modes of failure.

5A—Calculation of the Panel's Local Buckling:

In panels stiffened by triangular pockets, the pockets are triangularplates subjected to combined loads in the plane. In order to calculatethe buckling flow capacity and reserve factors under pure loading of thecomplete stiffened panel, a method based on a finite element model (FEM)is used.

This section provides a calculation of pocket buckling flow capacity forisosceles triangular pockets: the base of the triangle can vary betweenall values, and, in the present non-limitative example, between 45° and70°. The flow calculation is carried out with two types of boundaryconditions: simply supported and clamped

The applied stresses to be taken into account for the calculation of thereserve factor are solely the stresses affecting the skin, anddetermined in the section, described above, of the applied stresscalculation.

The entry data for this section are:

-   -   geometric data (base of the triangle, isosceles angle, thickness        of the skin)    -   material data (linear (E, ν) and non-linear (F_(cy), F_(tu), e        %, n_(c)))        -   uniquely isotropic material        -   the plastic buckling flow capacity values are only pertinent            up to F_(cy)    -   Boundary conditions: simply supported or clamped    -   Loads applied to the skin

It should be noted that all the external flows used in this section areskin flows and do not correspond with a complete loading of thestiffened panel.

In addition, the height of the triangle (h) used as reference length forthe buckling calculation is reduced to the semi-thickness of stiffenerwebs.

In this section, the formula used for the height of the triangle is:h=h_red

${h\_ red} = {{\frac{a\_ red}{2} \cdot \tan}\; \theta}$${a\_ red} = {a - {b \cdot \left\lbrack {\frac{1}{\tan \; \theta} + \frac{1}{\sin \; \theta}} \right\rbrack}}$

The exit data are the following:

-   -   Pocket buckling capacity.    -   Reserve factor

This section is aimed at calculating the flow capacities for isoscelestriangular plates.

It includes two parts: 1/A calculation of capacity values for simplysupported triangular plates (part 5A.5), 2/A calculation of capacityvalues for clamped triangular plates (part 5A.6).

The two following parts follow the same approach: firstly a calculationof capacity values for triangular plates subject to cases of pureloading (compression according to X, compression according to Y andshear load), then a calculation of interaction curves between the threecases of pure loading.

5A.1 Calculation Principle

The cases of pure loading envisaged are presented in FIG. 12.

Several analytical formula methods for local buckling of triangles existin written documentation. The comparison of these methods shows thatthere is a large difference between the previously cited bucklingstresses. Moreover, some of the parameters used in these methods arederived from calculations by finite elements, tests are often empiricaland certain methods provide absolutely no data for angles other thanthose of 60°.

The development of a complete theory of this problem being somewhatlengthy, the method such as is herein described, which isnon-limitative, implements a method based entirely on a Finite ElementModel (FEM):

-   -   Creating an FEM parametric model of a triangular plate        (parameters: base angle, thickness, height of triangle, boundary        conditions),    -   Testing various combinations to obtain linear results of        buckling,    -   Obtaining the parameters which will be used in an analytical        formula (coefficients K).

The induced plasticity effects must also be taken into account in thecalculation of capacity values. The applied loads are either simpleloads or combinations of these simple loads.

5A.2 Case of Pure Loading

Interaction curves are defined as follows. Six finite element models oftriangular plates were created with angles of between 45° and 70° in thepresent non-limitative example which concerns angles of the triangle. Inthis section, the isosceles angle (θ) is defined as the base angle ofthe isosceles triangle (see FIG. 18). For each isosceles angle and foreach case of pure loading, the study is organised in three points:

1/Calculation by Finite Element Model (FEM)

Linear calculations of local buckling of triangles by finite elementmodel (FEM) of a known type were carried out to determine the wrinkleflow capacity (without plastic correction) for various thicknesses, andtherefore various stiffnesses of a plate. We note that the first modeobserved always presents a single buckle (a single lump).

2/Tracing the Curve of Buckling Flow Capacity According to D/h²

In general, in written documentation, the buckling flow capacity isexpressed as follows:

$N_{crit} = {K \cdot \frac{D}{h^{2}}}$

K is a constant,

D the stiffness of the plate:

${D = \frac{E \cdot t^{3}}{12\left( {1 - v^{2}} \right)}},$

h is the height of the triangle:

$h = {{\frac{a}{2} \cdot \tan}\; {\theta.}}$

But this study demonstrates that, in the case of triangular plates, andfor the small values of the ratio

$\frac{D}{h^{2}},$

an expression of the buckling flow capacity using a first degreeequation in

$\frac{D}{h^{2}}$

is not pertinent. Better results are obtained with a second degreeequation in the following form:

$\begin{matrix}{N_{crit} = {{K_{1}\left( \frac{D}{h^{2}} \right)}^{2} + {{K_{2} \cdot \frac{D}{h^{2}}}\left( {{elastic}\mspace{14mu} {value}} \right)}}} & {{Equation}\mspace{14mu} 5\text{-}1}\end{matrix}$

Constants K₁ and K₂ depend on the angle and the load case beingconsidered. Therefore, for each case and each angle, a value for the K₁and K₂ constants is obtained.

3/ Tracing the Evolution of K₁ and K₂ According to the Base Angle of theIsosceles Triangle

K₁ and K₂ are traced according to the angle and an interpolation iscarried out to determine a polynomial equation which makes it possibleto calculate these constants of any angle between 45° and 70°. FIG. 19illustrates a linear or quadratic interpolation for the K coefficients.It is clear that this function also makes it possible to extrapolatevalues outside of, but close to the domain going from 45 to 70°. Thus,by knowing the isosceles angle and the boundary conditions, it ispossible to directly calculate the buckling flow capacity of thetriangular plate being studied.

5A.3 Case of Combined Loading

In this case the following hypothesis is used: if some components of thecombined load are under tension, these components are reduced to zero(are not taken into account for the calculation). It is, in effect,conservative to consider that the components under tension have noaffect with regards to the buckling flow being studied, and do notimprove the buckling stress on the plate. For example, if N_(x)^(app)=+200 N/mm (which shows a tension) and N_(s) ^(app)=300 N/mm, thecombined load capacity is reduced to the pure shear load capacity.

The presentation of the envisaged loading cases is illustrated in FIG.21. In this section, three finite element models were used: threeisosceles triangular plates with angles equal to 45°, 60° and 70°. Foreach angle, the study is organised in two points:

1/Calculation by Finite Element Model (FEM)

For all the combinations presented below, linear calculations by finiteelement method were carried out to determine the eigenvalue of bucklingcorresponding to different distributions of external loads.

We can see that the first mode observed always presents a singleblister.

It appears that the interaction curve depends little on the value of

$\frac{D}{h^{2\;}}.$

2/Tracing the Interaction Curves

The interaction curve is traced for each angle and each combination ofloads. Next, the various curves are approximated with classical curvesfor which the calculation takes the following form:

R ₁ ^(A) +R ₂ ^(B)=1

with

${R_{i} = \frac{N_{i}^{app}}{N_{i}^{crit}}},{i = {cX}},{{cY}\mspace{14mu} {or}\mspace{14mu} {s.}}$

The results and the choices made have shown that the equations ofinteraction curves do not depend on the base angle of the isoscelestriangle and are therefore compatible and can be unified by a singleequation covering all the combinations in the following form:

R _(cX) ^(A) +R _(cY) ^(B) +R _(s) ^(C)=1

Based on this equation, to determine the reserve factors, we can solvethe following equation:

${\left( \frac{R_{cy}}{R} \right)^{A} + \left( \frac{R_{cX}}{R} \right)^{B} + \left( \frac{R_{s}}{R} \right)^{C}} = 1$

with

$R = {\frac{N_{cY}^{app}}{N_{{cYcomb}\;}^{crit}} = {\frac{N_{cX}^{app}}{N_{cXcomb}^{crit}} = {\frac{N_{s}^{app}}{N_{scomb}^{crit}} = \frac{1}{RF}}}}$

5A.4 Plasticity Correction Factor

Obtaining a plasticity correction for cases of pure loading, accordingto the isosceles angle and the boundary conditions is very complex. Infact, for triangular plates, deflection functions are complex and giverise to numerous digital integration problems.

As a result, it was decided to use a conservative η factor, based on theNACA Report 898 (“A Unified Theory of Plastic Buckling of Columns andPlates”, July 1947).

This factor is defined for all cases of loading (pure and combined) withthe exception of shear load, by:

$\eta_{5} = \frac{E_{{ta}\; n}}{E_{c}}$

And for cases of pure shear load, by:

$\eta_{6} = {\frac{\left( {1 + v_{e}} \right)}{\left( {1 + v} \right)} \cdot \frac{E_{{se}\; c}}{E_{c}}}$

The correction is calculated by using the equivalent elastic stress ofVon Mises:

σ_(VM)=√{square root over (σ_(x) _(_) _(comb) ^(crit 2)+σ_(y) _(_)_(comb) ^(crit 2)−σ_(x) _(_) _(comb) ^(crit)·σ_(y) _(_) _(comb)^(crit)+3·τ_(xy) _(_) _(comb) ^(crit 2))}

Therefore, the corrected stress capacities can be calculated:

σ_(x) _(_) _(comb) ^(plast)=η·σ_(x) _(_) _(comb) ^(crit)

σ_(y) _(_) _(comb) ^(plast)=η·σ_(y) _(_) _(comb) ^(crit)

σ_(xy) _(_) _(comb) ^(plast)=η·τ_(xy) _(_) _(comb) ^(crit)

For cases of pure loading (compression according to X, compressionaccording to Y or shear load), the plasticity correction is also appliedto the Von Mises stress, therefore for cases of pure shear loading, thecorrected stress is: √{square root over (3)}·τ_(xy).

5A.5 Isosceles Triangular Simply Supported Plates

Case of Pure Loading

FIGS. 26 and 27 show the evolution of the K₁ and K₂ constants accordingto the isosceles triangle. The equations of these curves are (with θ indegrees):

K_(1cX)=−0.0000002417·θ⁴+0.0000504863·θ³−0.0039782194·θ²+0.1393226958·θ−1.8379213492

K _(1cY)c=−0.0000007200·θ⁴+0.0001511407·θ³−0.0119247778·θ²+0.4177844180·θ−5.4796530159

K_(1s)=−0.0000018083·θ⁴+0.0003804181·θ³−0.0300743972·θ²+1.0554840265·θ−13.8695053175

K _(2cX)=0.0029565·θ³−0.4291321·θ²+21.1697836·θ−291.6730902

K _(2cY)=0.0068664·θ³−1.0113413·θ²+51.3462358·θ−852.1945224

K _(2s)=0.013637·θ³−2.017207·θ²+102.120039·θ−1674.287384

Whether this is for K₁ or K₂, their values under pure compression X andpure Y are equal for an isosceles angle of 60°. The intersection pointat 60° is proof of the isotropic behaviour of the structure stiffened bytriangular pockets at 60°, in terms of local buckling in the skin.

Case of Combined Loading

In the case of combined loading, an analysis is carried out by finiteelement model of the linear calculation of buckling for simply supportedtriangular plates. We therefore choose a conservative interaction curve,close to the calculated interaction curves, but in a simple formula,which becomes the curve used in the method herein described. Itsequation is: R₁ ^(A)+R₂ ^(B)=1. With

${R_{i} = \frac{N_{i}^{app}}{N_{i}^{crit}}},$

i=cX, cY or s.

Interaction Compression X+Compression Y (Case 1)

In this case of loading, for angles between 45° and 70°, we chose aconservative interaction curve. That is to say a curve which declares aninteraction to a value lower than the sum of compressions R_(cX) on Xand R_(cY) on Y, in respect of all the interaction curves calculated forthe angle values between 45° and 70°. This curve is defined by thefollowing equation: R_(cX)+R_(cY)=1

Interaction Compression X+Shear Load (Case 2)

In this case of loading, for angles between 45° and 70°, we chose aconservative interaction curve, in respect of the different interactioncurves according to angles between 45° and 70°, defined by the followingequation: R_(cX)+R_(s) ^(3/2)=1

Interaction Compression Y+Shear Load (Case 3)

For angles between 45° and 70°, in order to determine the reserve factorin the case of a combined compression loading according to Y and ofshear load, we chose a conservative equation of the following formula:R_(cY)+R_(s) ²=1. In order to arrive at a single equation covering allcases of loading, we chose to use another, even more conservative curveR_(cY)+R_(s) ^(3/2)=1

Interaction Compression X+Compression Y+Shear Load (Case 4)

The equation chosen for these cases of loading is: R_(cX)+R_(cY)+R_(s)^(3/2)=1. This unique equation is used for all cases of combined loading

5A.6 Clamped Isosceles Triangular Plates

Case of Pure Loading

FIGS. 28 and 29 show the evolution of the K₁ and K₂ constants accordingto the isosceles triangle. The equations of these curves are (with θ indegrees):

K_(1cX)=−0.0000018547·θ⁴+0.0003940252·θ³−0.0314632778·θ²+1.1143937831·θ−14.8040153968

K_(1cY)=−0.0000027267·θ⁴+0.0005734489·θ³−0.0453489667·θ²+1.5921323016·θ−20.9299676191

K_(1s)=−0.0000069990·θ⁴+0.0014822211·θ³−0.1179080417·θ²+4.1617623127·θ−54.9899559524

K _(2cX)=0.0110488·θ³−1.6258419·θ²+81.8278420·0−1254.6580819

K _(2cY)=0.0158563·θ³−2.3439723·θ²+119.5038876−1970.9532998

K _(2s)=0.0252562·θ³−3.7563673·θ²+191.0642156·θ−3113.4527806

Case of Combined Loading

In the case of combined loading, an analysis is carried out by finiteelement model of the linear calculation of buckling for clampedtriangular plates. We therefore chose a conservative interaction curve,close to the calculated interaction curves, but in a simple formula,which becomes the curve used in the method herein described. Itsequation is: R₁ ^(A)+R₂ ^(B)=1 with

${R_{i} = \frac{N_{i}^{app}}{N_{i}^{crit}}},$

i=cX, cY or s.

Interaction Compression X+Compression Y (Case 1)

In this case of loading, for angles between 45° and 70°, we chose aconservative interaction curve, in respect of the different interactioncurves according to angles between 45° and 70°, defined by the followingequation: R_(cX)+R_(cY)=1

Interaction Compression X+Shear Load (Case 2)

In this case of loading, for angles between 45° and 70°, we chose aconservative interaction curve, in respect of the different interactioncurves according to angles between 45° and 70°, defined by the followingequation: R_(cX)+R_(s) ^(3/2)=1

Interaction Compression Y+Shear Load (Case 3)

For angles of between 45° and 70°, to determine the reserve factor inthe case of a combined compression load according to Y and of shearload, we chose a conservative equation of the following formula:R_(cY)+R_(s) ²=1. In order to arrive at a single equation covering allcases of loading, we chose to use another, even more conservative curveR_(cY)+R_(s) ^(3/2)=1

Interaction: Compression X+Compression Y+Shear Load (Case 4)

The equation, used for this case of loading, is R_(cX)+R_(cY)+R_(s)^(3/2)=1. This unique equation is used for all cases of combinedloading.

5B Calculation of Local Buckling of the Stiffener:

This modulus calculates the buckling stress and the reserve factor ofthe stiffener web, considered as a rectangular panel with diverseboundary conditions to be defined by the user.

The applied stresses to take into account for the calculations ofreserve factor are uniquely the stresses in the stiffener webs, derivedfrom the modulus of calculation of applied stresses.

On the stiffener grid, one or several types of stiffener webs arecompression loaded. Because of this the compression stress capacity mustbe calculated.

The entry data for this module are:

-   -   Geometric data: dimensions of stiffener webs (length, height,        thickness),    -   Material data (linear (E,ν) and non-linear (F_(cy), F_(tu),        ε_(ult), n_(c))).

In the present example we are only considering an isotropic material,

-   -   boundary conditions (four are available),    -   Loads applied to the stiffener webs.

The exit data are the buckling capacity of the stiffener web and areserve factor. The buckling stress capacity of the stiffener web is(see FIG. 13 for the notation conventions):

$\begin{matrix}{\sigma_{blade}^{crit} = {k_{c} \cdot \frac{\eta \cdot E_{c} \cdot \pi^{2}}{12 \cdot \left( {1 - v_{e}^{2}} \right)} \cdot \left( \frac{b}{d} \right)^{2}}} & {{Equation}\mspace{14mu} 5\text{-}2}\end{matrix}$

-   -   With:        -   b: thickness of the stiffener web        -   d: height of the stiffener web        -   L_(b): length of the stiffener web        -   E_(c): Young's modulus in compression        -   ν_(e): Poisson coefficient in the elastic domain        -   k_(c): Local buckling factor (dependent on boundary            conditions and geometry)        -   η: Plasticity correction factor

Note: the length of a stiffener web is given by

-   -   (L_(b))=a (for stiffener webs in the X direction)

$\left( L_{b} \right) = {\frac{a}{2} \cdot \sqrt{1 + {\tan^{2}\theta}}}$

(for stiffener webs in transversal directions)

Numerous boundary conditions can be applied on the stiffener webaccording to the surrounding structure. (see FIG. 14). We note that ifLb/d is greater than the value of Lim, then k_(c) is worth k_(c)infinitively. The recommended conservative buckling factor forcalculations based on numerous finite element analyses case 2 (2 clampededges—one simply supported edge—one free edge)

According to the boundary conditions cited above and according to table1 of plastic correction factors for rectangular plates, the plasticitycorrection factor used in this case is:

$\eta = {\eta_{1} = {\frac{E_{s\; {ec}}}{E_{c}} \cdot \frac{1 - v_{e}^{2}}{1 - v^{2}}}}$

The formula for reserve factor calculation for buckling of the stiffenerweb is valid for all types of stiffener webs used in the present panelstiffened by triangular pockets (0°, +θ, −θ):

${RF}_{buck}^{blade} = \frac{\sigma_{blade}^{crit}}{\sigma_{blade}^{app}}$

The following example is based on the same geometry as was used in theprevious sections. The geometry of stiffeners is: b=2.5 mm, and d=37.36mm. The length of stiffener webs is (L_(b)): L_(b)=a=198 mm forstiffener webs in the X direction and

$L_{b} = {{\frac{a}{2} \cdot \sqrt{1 + {\tan^{2}\theta}}} = {186.82\mspace{14mu} {mm}}}$

for transversal stiffener webs. The boundary conditions used are: 2sides clamped—1 side simply supported—1 side free.

Thus:

$k_{c\; \_ \; 0{^\circ}} = {{{4.143 \cdot \left( \frac{d}{L_{b}} \right)^{2}} + 0.384} = 0.5315}$

for stiffener webs in the X direction and

$k_{c\; \_ \; {\theta{^\circ}}} = {{{4.143 \cdot \left( \frac{d}{L_{b}} \right)^{2}} + 0.384} = 0.5497}$

for transversal stiffener webs

And the buckling load for each stiffener is:

The plasticity correction factor at stiffener web buckling is:

$\eta = {\eta_{1} = {{\frac{E_{\sec}}{E_{c}} \cdot \frac{1 - v_{e}^{2}}{1 - v^{2}}} = {1\mspace{14mu} ({elastic})}}}$

The loads applied in the stiffeners are:

-   -   σ_(0°)=−101.14 MPa    -   σ_(+θ)=44.25 MPa    -   σ_(−θ)=−135.07 MPa

The results of reserve fracture calculation are the following:

Step 6—Calculation of General Instability:

This step provides data on buckling flow capacity for a flat panelstiffened by triangular pockets, in the conditions of pure or combinedloading.

The formulae are based on the buckling of orthotropic plates. Two orfour boundary conditions are possible according to the loading case (4simply-supported edges, 4 clamped edges, 2 loaded simply supported edgesand 2 lateral clamped edges, 2 loaded clamped edges and 2 simplysupported lateral edges). The applied flows, to take into account forreserve factor calculation, are the external flows of the panelstiffened by triangular pockets which are the entry data.

The entry data are the following:

-   -   Geometric data:        -   L_(x): length of the panel equivalent to the grid        -   L_(y): width of the panel equivalent to the grid,        -   t_(s): thickness of the skin,        -   t_(g): thickness of the panel equivalent to the grid,    -   Data on the material:        -   E_(x) ^(s), E_(y) ^(s): Young's modulus of the skin,        -   G_(xy) ^(s): shear modulus of the skin,        -   ν_(xy) ^(s), ν_(yx) ^(s): Poisson coefficient of the skin,        -   E_(x) ^(g), E_(y) ^(g): Young's modulus of the grid,        -   G_(xy) ^(g): shear modulus of the grid,        -   ν_(xy) ^(g), ν_(yx) ^(g): Poisson coefficient of the grid,    -   loads applied to the structure (N_(x) ⁰, N_(y) ⁰, N_(xy) ⁰,        p_(z))    -   boundary conditions (2 or 4 are possible according to the type        of loading)

The exit data are:

-   -   N_(x) ^(c), N_(y) ^(c), N_(xy) ^(c): buckling flow capacity,    -   N_(x) ^(c) _(comb), N_(y) ^(c) _(comb), N_(xy) ^(c) _(comb):        combined buckling flow capacity,    -   Reserve factors

We use the Kirchhoff hypothesis: plane sections remain principally planefollowing strain. The grid (of stiffeners) is modelled here by anequivalent panel. The skin and the panel equivalent to the grid areconsidered as plates of an orthotropic nature.

The material parameters verify the following relation:

$\frac{v_{xy}^{i}}{E_{x}^{i}} = \frac{v_{yx}^{i}}{E_{y}^{i}}$

The conventions of flow and moments are illustrated by FIG. 21.

6.1.1 Displacements

The vector {right arrow over (U)} represents the displacement of a pointM(x,y) of the median surface: {right arrow over(U)}=[u,v,w]=u(x,y){right arrow over (x)}+v(x,y){right arrow over(y)}+w(x,y){right arrow over (z)}

The different variables do not depend on z because a plane state ofstresses has been envisaged (σ_(zz)=0).

6.1.2 Strains

The general expression of strains in a section of the plate situated ata z distance from the median axis is:

$\begin{matrix}\left\{ \begin{matrix}{ɛ_{xx} = {ɛ_{xx}^{0} + {z.\kappa_{xx}}}} \\{ɛ_{yy} = {ɛ_{yy}^{0} + {z.\kappa_{yy}}}} \\{ɛ_{xy} = {ɛ_{xy}^{0} + {z.\kappa_{xy}}}}\end{matrix} \right. & {{Equation}\mspace{14mu} 6\text{-}1}\end{matrix}$

With:

$\begin{matrix}\left\{ \begin{matrix}{ɛ_{xx}^{0} = {u_{,x} + {\frac{1}{2}\left( w_{,x} \right)^{2}}}} \\{ɛ_{yy}^{0} = {v_{,y} + \frac{w}{R} + {\frac{1}{2}\left( w_{,y} \right)^{2}}}} \\{ɛ_{xy}^{0} = {{\frac{1}{2}\left( {u_{,y} + v_{,x}} \right)} + {\frac{1}{2}{w_{,x}.w_{,y}}}}}\end{matrix} \right. & {{Equation}\mspace{14mu} 6\text{-}2}\end{matrix}$

The terms ε_(xx) ⁰, ε_(yy) ⁰, and ε_(xy) ⁰ represent the contribution instrain in the plane of the plate. The terms ½(w_(,x))², ½(w_(,y))², and½w_(,x)·w_(,y) represent the non-linear contribution in strain in theplane of the plate. The term R represents the radius of the shell, buthere we are considering a plane plate, therefore

$\frac{1}{R} = 0.$

$\begin{matrix}\left\{ \begin{matrix}{\kappa_{xx} = {- w_{,x^{2}}}} \\{\kappa_{yy} = {- w_{,y^{2}}}} \\{\kappa_{xy} = {- w_{,{xy}}}}\end{matrix} \right. & {{Equation}\mspace{14mu} 6\text{-}3}\end{matrix}$

The terms z·κ_(xx), z·κ_(yy), and z·κ_(xy) represent the contribution instrain due to the change of the plate curve (z is the distance from themedian axis of the plate).

6.1.3 Behaviour Laws

The sink and the panel equivalent to the stiffeners are considered asorthotropic plates. Because of this the relations between stresses andstrains are:

$\begin{matrix}{\begin{pmatrix}\sigma_{xx}^{i} \\\sigma_{yy}^{i} \\\sigma_{xy}^{i}\end{pmatrix} = {\begin{pmatrix}\frac{E_{x}^{i}}{1 - {v_{xy}^{i}v_{yx}^{i}}} & \frac{v_{xy}^{i}E_{y}^{i}}{1 - {v_{xy}^{i}v_{yx}^{i}}} & 0 \\\frac{v_{yx}^{i}E_{x}^{i}}{1 - {v_{xy}^{i}v_{yx}^{i}}} & \frac{E_{y}^{i}}{1 - {v_{xy}^{i}v_{yx}^{i}}} & 0 \\0 & 0 & {2G_{xy}^{i}}\end{pmatrix}\begin{pmatrix}ɛ_{xx} \\ɛ_{yy} \\ɛ_{xy}\end{pmatrix}}} & {{Equation}\mspace{14mu} 6\text{-}4}\end{matrix}$

with i=(s, g) (indices s for the values relative to the skin and indicesg for the values relative to the stiffener grid).

6.1.4 Flow and Moments

The expressions of flow and moments by unit of length are:

$\begin{matrix}{{N_{\alpha\beta} = {\int_{h}{\sigma_{\alpha\beta}{dz}}}}{M_{\alpha\beta} = {\int_{h}{{\sigma \ }_{\alpha\beta}.{zdz}}}}} & {{Equation}\mspace{14mu} 6\text{-}5}\end{matrix}$

with (α,β)=(x,y).

Flow:

$N_{\alpha\beta} = {{\int_{h}{\sigma_{\alpha\beta}{dz}}} = {{\int_{- \frac{h}{2}}^{{- \frac{h}{2}} + t_{s}}{\sigma_{\alpha\beta}^{s}{dz}}} + {\int_{{- \frac{h}{2}} + t_{s}}^{\frac{h}{2}}{\sigma_{\alpha\beta}^{g}{dz}}}}}$

By using Equation 6-5 and the relation h=t_(s)+t_(g) we find:

$\begin{matrix}{N_{xx} = {{\left( {\frac{E_{x}^{s}t_{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}} + \frac{E_{x}^{g}t_{g}}{1 - {v_{xy}^{g}v_{yx}^{g}}}} \right)ɛ_{xx}^{0}} + {\left( {\frac{v_{xy}^{s}E_{y}^{s}t_{s}}{1 - {v_{xy}^{g}v_{yx}^{g}}} + \frac{v_{xy}^{g}E_{y}^{g}t_{g}}{1 - {v_{xy}^{g}v_{yx}^{g}}}} \right)ɛ_{yy}^{0}} + {\left\lbrack {\frac{1}{2}t_{s}{t_{g}\left( {\frac{E_{x}^{g}}{1 - {v_{xy}^{g}v_{yx}^{g}}} - \frac{E_{x}^{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}}} \right)}} \right\rbrack \kappa_{xx}} + {\left\lbrack {\frac{1}{2}t_{s}{t_{g}\left( {\frac{v_{xy}^{g}E_{y}^{g}}{1 - {v_{xy}^{g}v_{yx}^{g}}} - \frac{v_{xy}^{s}E_{y}^{s}}{1 - {v_{xy}^{g}v_{yx}^{g}}}} \right)}} \right\rbrack \kappa_{yy}}}} & {{Equation}\mspace{14mu} 6\text{-}6} \\{N_{yy} = {{\left( {\frac{v_{yx}^{s}E_{x}^{s}t_{s}}{1 - {v_{xy}^{g}v_{yx}^{g}}} + \frac{v_{yx}^{g}E_{x}^{g}t_{g}}{1 - {v_{xy}^{g}v_{yx}^{g}}}} \right)ɛ_{xx}^{0}} + {\left( {\frac{E_{y}^{s}t_{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}} + \frac{E_{y}^{g}t_{g}}{1 - {v_{xy}^{g}v_{yx}^{g}}}} \right)ɛ_{yy}^{0}} + {\left\lbrack {\frac{1}{2}t_{s}{t_{g}\left( {\frac{v_{yx}^{g}E_{x}^{g}}{1 - {v_{xy}^{g}v_{yx}^{g}}} - \frac{v_{yx}^{s}E_{x}^{s}}{1 - {v_{xy}^{g}v_{yx}^{g}}}} \right)}} \right\rbrack \kappa_{xx}} + {\quad{\left\lbrack {\frac{1}{2}t_{s}{t_{g}\left( {{\frac{1}{2}t_{s}t_{g}\frac{E_{y}^{g}}{1 - {v_{xy}^{g}v_{yx}^{g}}}} - \frac{E_{y}^{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}}} \right)}} \right\rbrack \kappa_{yy}}}}} & \; \\{\mspace{79mu} {N_{xy} = {{2\left( {{G_{xy}^{s}t_{s}} + {G_{xy}^{g}t_{g}}} \right)ɛ_{xy}^{0}} + {\left\lbrack {t_{s}{t_{g}\left( {G_{xy}^{g} - G_{xy}^{s}} \right)}} \right\rbrack \kappa_{xy}}}}} & \;\end{matrix}$

Moments by Unit of Length:

$M_{\alpha\beta} = {{\int_{h}{\sigma_{\alpha\beta}{zdz}}} = {{\int_{- \frac{h}{2}}^{{- \frac{h}{2}} + t_{s}}{\sigma_{\alpha\beta}^{s}{zdz}}} + {\int_{{- \frac{h}{2}} + t_{s}}^{\frac{h}{2}}{\sigma_{\alpha\beta}^{g}{zdz}}}}}$

By using Equation 6-5 and the relation h=t_(s)+t_(g) we find:

$\begin{matrix}{{- M_{xx}} = {{\left\lbrack {\frac{1}{2}t_{s}{t_{g}\left( {\frac{E_{x}^{g}}{1 - {v_{xy}^{g}v_{yx}^{g}}} - \frac{E_{x}^{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}}} \right)}} \right\rbrack ɛ_{xx}^{0}} + {\quad{{\left\lbrack {\frac{1}{2}t_{s}{t_{g}\left( {\frac{v_{xy}^{g}E_{y}^{g}}{1 - {v_{xy}^{g}v_{yx}^{g}}} - \frac{v_{xy}^{s}E_{y}^{s}}{1 - {v_{xy}^{g}v_{yx}^{g}}}} \right)}} \right\rbrack ɛ_{yy}^{0}} + {\quad{{\left\lbrack {\frac{1}{12}\left( {\frac{E_{x}^{s}{t_{s}\left( {t_{s}^{2} + {3t_{g}^{2}}} \right)}}{1 - {v_{xy}^{s}v_{yx}^{s}}} + \frac{E_{x}^{g}{t_{g}\left( {t_{g}^{2} + {3t_{s}^{2}}} \right)}}{1 - {v_{xy}^{g}v_{yx}^{g}}}} \right)} \right\rbrack \kappa_{xx}} + {\quad{\left\lbrack {\frac{1}{12}\left( {\frac{v_{xy}^{s}E_{y}^{s}{t_{s}\left( {t_{s}^{2} + {3t_{g}^{2}}} \right)}}{1 - {v_{xy}^{g}v_{yx}^{g}}} + \frac{v_{xy}^{g}E_{y}^{g}{t_{g}\left( {t_{g}^{2} + {3t_{s}^{2}}} \right)}}{1 - {v_{xy}^{g}v_{yx}^{g}}}} \right)} \right\rbrack \kappa_{yy}}}}}}}}} & {{Equation}\mspace{14mu} 6\text{-}7} \\{M_{yy} = {{\left\lbrack {\frac{1}{2}t_{s}{t_{g}\left( {\frac{v_{yx}^{g}E_{x}^{g}}{1 - {v_{xy}^{g}v_{yx}^{g}}} - \frac{v_{yx}^{s}E_{x}^{s}}{1 - {v_{xy}^{g}v_{yx}^{g}}}} \right)}} \right\rbrack ɛ_{xx}^{0}} + {\quad{{\left\lbrack {\frac{1}{2}t_{s}{t_{g}\left( {\frac{E_{y}^{g}}{1 - {v_{xy}^{g}v_{yx}^{g}}} - \frac{E_{y}^{s}}{1 - {v_{xy}^{s}v_{yx}^{s}}}} \right)}} \right\rbrack ɛ_{yy}^{0}} + {\quad{{\left\lbrack {\frac{1}{12}\left( {\frac{\quad{v_{yx}^{s}E_{x}^{s}{t_{s}\left( {t_{s}^{2} + {3t_{g}^{2}}} \right)}}}{1 - {v_{xy}^{g}v_{yx}^{g}}} + \frac{v_{yx}^{g}E_{x}^{g}{t_{g}\left( {t_{g}^{2} + {3t_{s}^{2}}} \right)}}{1 - {v_{xy}^{g}v_{yx}^{g}}}} \right)} \right\rbrack \kappa_{xx}} + {\quad{\left\lbrack {\frac{1}{12}\left( {\frac{E_{y}^{s}{t_{s}\left( {t_{s}^{2} + {3t_{g}^{2}}} \right)}}{1 - {v_{xy}^{s}v_{yx}^{s}}} + \frac{E_{y}^{g}{t_{g}\left( {t_{g}^{2} + {3t_{s}^{2}}} \right)}}{1 - {v_{xy}^{g}v_{yx}^{g}}}} \right)} \right\rbrack \kappa_{yy}}}}}}}}} & \; \\{{- M_{xy}} = {{\left\lbrack {t_{s}{t_{g}\left( {G_{xy}^{g} - G_{xy}^{s}} \right)}} \right\rbrack ɛ_{xy}^{0}} + {\quad{\left\lbrack {\frac{1}{6}\left( {{G_{xy}^{s}{t_{s}\left( {t_{s}^{2} + {3t_{g}^{2}}} \right)}} + {G_{xy}^{g}{t_{g}\left( {t_{g}^{2} + {3t_{s}^{2}}} \right)}}} \right)} \right\rbrack \kappa_{xy}}}}} & \;\end{matrix}$

Once the general behaviour law (between flow and moments on one hand andstrains on the other), is obtained:

Matrices A, B and C are symmetrical.

6.2 Balance Equations

General balance equations of an element of the panel (or shell) aregiven by the following expressions, linking flows, moments and surfacestrength density:

$\begin{matrix}\left\{ \begin{matrix}{{N_{{xx},x} + N_{{xy},y} + p_{x}} = 0} \\{{N_{{yy},y} + N_{{yx},x} + p_{y}} = 0} \\{\left( {N_{xx}w_{,x}} \right)_{,x} + \left( {N_{yy}w_{,y}} \right)_{,y} + \left( {N_{xy}w_{,y}} \right)_{,x} + \left( {N_{yx}w_{,x}} \right)_{,y} - \frac{N_{yy}}{R} -} \\{{M_{{xx},x^{2}} + M_{{yy},y^{2}} - {2M_{{xy},{xy}}} + p_{z}} = 0}\end{matrix} \right. & {{Equation}\mspace{14mu} 6\text{-}9}\end{matrix}$

where {right arrow over (f)}=p_(x){right arrow over (x)}+p_(y){rightarrow over (y)}+p_(z){right arrow over (z)} is the surface strengthdensity acting on the shell element. The surface strength density onlyacts along the Z radial direction, and not in the other directions.Therefore, p_(x)=p_(y)=0. Furthermore, in this instance we areconsidering the case of a flat plate, therefore

$\frac{1}{R} = 0.$

Because of this we obtain simplified balance equations:

$\begin{matrix}\left\{ {\quad\begin{matrix}{{N_{{xx},x} + N_{{xy},y}} = 0} \\{{N_{{yy},y} + N_{{yx},x}} = 0} \\{{{N_{xx}w_{,x^{2}}} + {N_{yy}w_{,y^{2}}} + {2\; N_{xy}w_{,{xy}}} + p_{z}} = {M_{{xx},x^{2}} - M_{{yy},y^{2}} + {2\; M_{{xy},{xy}}}}}\end{matrix}} \right. & {{Equation}\mspace{14mu} 6\text{-}1}\end{matrix}$

6.3 For the general solution of these equations, we define the followingvectors

$\begin{matrix}{\lbrack ɛ\rbrack = {{\begin{bmatrix}ɛ_{xx} \\ɛ_{yy} \\ɛ_{xy}\end{bmatrix}\lbrack N\rbrack} = {{\begin{bmatrix}N_{xx} \\N_{yy} \\N_{xy}\end{bmatrix}\lbrack K\rbrack} = {{\begin{bmatrix}\kappa_{xx} \\\kappa_{yy} \\\kappa_{xy}\end{bmatrix}\lbrack W\rbrack} = {{\begin{bmatrix}w_{,{xx}} \\w_{,{yy}} \\w_{,{xy}}\end{bmatrix}\lbrack M\rbrack} = \begin{bmatrix}{- M_{xx}} \\M_{yy} \\{- M_{xy}}\end{bmatrix}}}}}} & {{Equation}\mspace{14mu} 6\text{-}11}\end{matrix}$

The usual loads applied on the plate are:

-   -   Uniform compression flow along the x axis: −N_(x) ⁰    -   Uniform compression flow along the y axis: −N_(y) ⁰    -   Uniform shear flow in the x-y plane: −N_(xy) ⁰    -   Uniform Pressure along the z axis: p_(z){right arrow over (z)}

Because of this, since the applied loads defined below are uniform, itcan be deduced that the two first equations of Equation 6-10 areverified. N_(xx,x)=N_(yy,y)=N_(xy,y)=N_(yx,x)=0.

Expression of Moments:

By using Equation 6-8 and Equation 6-11, the following relations arefound:

[N]=A·[ε]+B·[K]

[M]=B·[ε]+C·[K]   Equation 6-12

As the applied flows are uniform, the following relations are obtained:

[N] _(,x) ₂ =0

A·[ε] _(,x) ₂ +B·[ε] _(,x) ₂ =0

[ε]_(,x) ₂ =A ⁻¹ ·B·[W] _(,x) ₂

[N] _(,y) ₂ =0

A·[ε] _(,y) ₂ +B·[ε] _(,y) ₂ =0

[ε]_(,y) ₂ =A ⁻¹ ·B·[W] _(,y) ₂

[N] _(,xy)=0

A·[ε] _(,xy) +B·[ε] _(,xy)=0

[ε]_(,xy) =A ⁻¹ ·B·[W] _(,xy)   Equation 6-13

With the fact that: [W]=−[K].

We therefore have for moments:

[M] _(,x) ₂ =B·[ε] _(,x) ₂ +C·[K] _(,x) ₂ =( B·A ⁻¹ ·B−C )·[W] _(,x) ₂=−D·[W] _(,x) ₂

[M] _(,y) ₂ =B·[ε] _(,y) ₂ +C·[K] _(,y) ₂ =( B·A ⁻¹ ·B−C )·[W] _(,y) ₂=−D·[W] _(,y) ₂

[M] _(,xy) =B·[ε] _(,xy) +C·[K] _(,xy)=( B·A ⁻¹ ·B−C )·[W] _(,xy)=−D·[W] _(,xy)   Equation 6-14

with

D=C−B·A ⁻¹ ·B    Equation 6-15

D is the global stiffness matrix, and is symmetrical.

$\overset{\_}{\overset{\_}{D}} = \begin{pmatrix}D_{11} & D_{12} & 0 \\D_{12} & D_{22} & 0 \\0 & 0 & D_{33}\end{pmatrix}$

Derivations of the moments of the Equation 6-10 are therefore obtained:

M _(xx,x) ₂ =D ₁₁ w _(,x) ₄ +D ₁₂ w _(,x) ₂ _(y) ₂

M _(yy,y) ₂ =D ₁₂ w _(,x) ₂ _(y) ₂ +D ₂₂ w _(,y) ₄

M _(xy,xy) =D ₃₃ w _(,x) ₂ _(y) ₂    Equation 6-16

The expression of Equation 6-10 in terms of displacements thereforegives the general differential equation:

−N _(x) ⁰ w _(,x) ₂ −N _(y) ⁰ w _(,y) ₂ −2N _(xy) ⁰ w _(,xy) +p _(z)=Ω₁w _(,y) ₄ +Ω₂ w _(,y) ₄ +Ω₃ w _(,x) ₂ _(y) ₂    Equation 6-17

with:

Ω₁ =D ₁₁

Ω₂ =D ₂₂

Ω₃=2·(D ₁₂ +D ₃₃)   Equation 6-18

In the following section, the panel stiffened by triangular pockets ismodelled with its three bending stiffeners (Ω1, Ω2 and Ω3) in order tocalculate the buckling flows in an orthotropic plate.

Here again, the following hypothesis is used in this case: if somecomponents of the combined load are in tension, these components are nottaken into account for the calculation. It is, in effect, conservativeto consider that the components in tension have no affect with regardsto the buckling flow being studied, and do not improve the bucklingstress on the plate.

6.4 Buckling Flow Capacity

6.4.1 Longitudinal Compression Flow (Compression According to X)

The plate is subjected to a uniform longitudinal compression flow(according to the X axis): −N_(x) ⁰. Because of this: N_(y) ⁰=N_(xy)⁰=p_(z)=0. The general differential equation (Equation 6-17) thereforeexpresses:

−N _(x) ⁰ w _(,x) ₂ =Ω₁ w _(,x) ₄ +Ω₂ w _(,y) ₄ +Ω₃ w _(,x) ₂ _(y) ₂   Equation 6-19

Firstly, we consider a simply supported plate (the boundary conditionsare generalised further on):

For x=0 and x=L_(x): w=0 and −M_(xx)=0

For y=0 and y=L_(y): w=0 and M_(yy)=0

The following expression for displacement w satisfies all the boundaryconditions detailed above:

$\begin{matrix}{{w\left( {x,y} \right)} = {{C_{mn}{\sin \left( \frac{m\; \pi \; x}{L_{x}} \right)}{\sin \left( \frac{n\; \pi \; y}{L_{y}} \right)}\left( {m,n} \right)} \in N^{2}}} & {{Equation}\mspace{14mu} 6\text{-}20}\end{matrix}$

The previous expression for w must satisfy the general differentialequation (Equation 6-19), therefore we obtain:

${N_{x}^{0}\left( \frac{m\; \pi}{L_{x}} \right)}^{2} = {{{\Omega_{1}\left( \frac{m\; \pi}{L_{x}} \right)}^{4} + {\Omega_{2}\left( \frac{n\; \pi}{L_{y}} \right)}^{4} + {{\Omega_{3}\left( \frac{m\; \pi}{L_{x}} \right)}^{2}\left( \frac{n\; \pi}{L_{y}} \right)^{2}\left( {m,n} \right)}} \in N^{2}}$

The minimum value of N⁰ corresponds to the value of flow capacity ofgeneral buckling N_(x) ^(c). We demonstrate that this value is:

$\begin{matrix}{N_{x}^{c} = {2\left( \frac{\pi}{L_{y}} \right)^{2}\left( {\sqrt{\Omega_{1}\Omega_{2}} + \frac{\Omega_{3}}{2}} \right)}} & {{Equation}\mspace{14mu} 6\text{-}21}\end{matrix}$

This formula can be generalised for different boundary conditions(loaded edges and simply supported or clamped lateral edges):

$\begin{matrix}{N_{x}^{c} = {{k_{c}\left( \frac{\pi}{L_{y}} \right)}^{2}\sqrt{\Omega_{1}\Omega_{2}}}} & {{Equation}\mspace{14mu} 6\text{-}22}\end{matrix}$

with

$k_{c} = {{h\left( \overset{\_}{\alpha} \right)} + {q \cdot \beta}}$$\overset{\_}{\alpha} = {\frac{L_{x}}{L_{y}}\sqrt[4]{\frac{\Omega_{2}}{\Omega_{1}}}}$$\beta = \frac{\Omega_{3}}{2\sqrt{\Omega_{1}\Omega_{2}}}$${h\left( \overset{\_}{\alpha} \right)} = \left\{ {{\begin{matrix}{{\left( \frac{1}{\overset{\_}{\alpha}} \right)^{2} + {\overset{\_}{\alpha}}^{2}},} & {{{si}\; \overset{\_}{\alpha}} \leq 1} \\{2,} & {\sin \mspace{14mu} {on}}\end{matrix}q} = 2} \right.$

FIG. 22 shows the value of h(a) according to different boundaryconditions (the case of four simply supported edges is the base curve).

6.4.2 Transversal Compression Flow (Compression According to Y)

The plate is subjected to a uniform transversal compression flow(according to the Y axis): −N_(y) ⁰. Because of this: N_(x) ⁰=N_(xy)⁰=p_(z)=0. The solution is the same as the one described in the previoussection.

The buckling capacity flow N is expressed by:

$\begin{matrix}{N_{y}^{c} = {{k_{c}\left( \frac{\pi}{L_{x}} \right)}^{2}\sqrt{\Omega_{2}\Omega_{1}}}} & {{Equation}\mspace{14mu} 6\text{-}23}\end{matrix}$

with

$k_{c} = {{h\left( \overset{\_}{\alpha} \right)} + {q \cdot \beta}}$$\overset{\_}{\alpha} = {\frac{L_{y}}{L_{x}}\sqrt[4]{\frac{\Omega_{1}}{\Omega_{2}}}}$$\beta = \frac{\Omega_{3}}{2\sqrt{\Omega_{2}\Omega_{1}}}$${h\left( \overset{\_}{\alpha} \right)} = \left\{ {{\begin{matrix}{{\left( \frac{1}{\overset{\_}{\alpha}} \right)^{2} + {\overset{\_}{\alpha}}^{2}},} & {{{si}\; \overset{\_}{\alpha}} \leq 1} \\{2,} & {\sin \mspace{14mu} {on}}\end{matrix}q} = 2} \right.$

6.4.3 Shear Flow

The plate is subjected to a uniform shear flow: −N_(xy) ⁰. Because ofthis: N_(x) ⁰=N_(y) ⁰=p_(z)=0. We note that in this paragraph, thefollowing formulae are only validated for Ly<Lx. In the opposite case,some terms must be exchanged: L_(x)

L_(y), and Ω₁

Ω₂. The buckling flow capacity N⁰ is expressed by:

$\begin{matrix}{N_{xy}^{c} = {{k_{s}\left( \frac{\pi}{L_{y}} \right)}^{2}\sqrt[4]{\Omega_{1}\Omega_{2}^{3}}}} & {{Equation}\mspace{14mu} 6\text{-}24}\end{matrix}$

With k_(s) obtained from the graph in FIG. 23 and the table in FIG. 24(for the case of simply supported edges), and in FIG. 25 (case of fourclamped edges) (source: S. G. Lekhnitskii: Anisotropic Plates. Gordonand Breach).

The entry data of the table and the graph are defined with:

$\overset{\_}{\alpha} = {\frac{L_{y}}{L_{x}}\sqrt[4]{\frac{\Omega_{1}}{\Omega_{2}}}}$$\beta = \frac{\Omega_{3}}{2\sqrt{\Omega_{2}\Omega_{1}}}$

6.4.4 Biaxial Compression Flow

The plate is subjected to combined loading: a uniform longitudinalcompression flow (according to the x axis) and a uniform transversalcompression flow (according to the y axis): −N_(x) _(comb) ⁰ and −N_(y)_(comb) ⁰ Because of this: N_(xy) ⁰=p_(z)=0. We define λ by: N_(y)_(comb) ⁰=λN_(x) _(comb) ⁰. The general differential equation isexpressed

−N _(x) _(comb) ⁰ w _(,x) ₂ −N _(y) _(comb) ⁰ w _(,y) ₂ =Ω₁ w _(,x) ₄+Ω₂ w _(,y) ₄ +Ω₃ w _(,x) ₂ _(y) ₂    Equation 6-25

Boundary Conditions Four Simply Supported Edges:

For x=0 and x=L_(x): w=0 and −M_(xx)=0

For y=0 and y=L_(y): w=0 and M_(yy)=0

Expression of Displacement:

The following expression of displacement w satisfies all the boundaryconditions detailed above:

$\begin{matrix}{{w\left( {x,y} \right)} = {{C_{mn}{\sin \left( \frac{m\; \pi \; x}{L_{x}} \right)}{\sin \left( \frac{n\; \pi \; y}{L_{y}} \right)}\left( {m,n} \right)} \in N^{2}}} & {{Equation}\mspace{14mu} 6\text{-}26}\end{matrix}$

Buckling flow capacity N_(x) _(comb) ^(c), N_(y) _(comb) ^(c)):

The previous expression for w must satisfy the general differentialequation (Equation 6-25), and we therefore obtain:

${{N_{x\; {comb}}^{0}\left( \frac{m\; \pi}{L_{x}} \right)}^{2} + {N_{y\; {comb}}^{0}\left( \frac{n\; \pi}{L_{y}} \right)}^{2}} = {{{\Omega_{1}\left( \frac{m\; \pi}{L_{x}} \right)}^{4} + {\Omega_{2}\left( \frac{n\; \pi}{L_{y}} \right)}^{4} + {{\Omega_{3}\left( \frac{m\; \pi}{L_{x}} \right)}^{2}\left( \frac{n\; \pi}{L_{y}} \right)^{2}\left( {m,n} \right)}} \in N^{2}}$

The expression of N_(x) _(comb) ⁰ according to A must satisfy:

$N_{x\; {comb}}^{0} = {{{\frac{\pi^{2}}{{L_{y}^{2}m^{2}} + {\lambda \; L_{x}^{2}n^{2}}}\left\lbrack {{{\Omega_{1}\left( \frac{L_{y}}{L_{x}} \right)}^{2}m^{4}} + {{\Omega_{2}\left( \frac{L_{x}}{L_{y}} \right)}^{2}n^{4}} + {\Omega_{3}m^{2}n^{2}}} \right\rbrack}\left( {m,n} \right)} \in N^{2}}$

thus, the buckling flow capacities are obtained:

$\begin{matrix}{{N_{x_{comb}}^{c} = {{Min}\left\{ {{\frac{\pi^{2}}{{L_{y}^{2}m^{2}} + {\lambda \; L_{x}^{2}n^{2}}}\left\lbrack {{{\Omega_{1}\left( \frac{L_{y}}{L_{x}} \right)}^{2}m^{4}} + {{\Omega_{2}\left( \frac{L_{x}}{L_{y}} \right)}^{2}n^{4}} + {\Omega_{3}m^{2}n^{2}}} \right\rbrack},{\left( {m,n} \right) \in N^{2}}} \right\}}}\mspace{20mu} {N_{y_{comb}}^{c} = {\lambda \; N_{x_{comb}}^{c}}}} & {{Equation}\mspace{14mu} 6\text{-}1}\end{matrix}$

Boundary Conditions: Four Clamped Edges:

For x=0 and x=L_(x):

$w = {{0\mspace{14mu} {and}\mspace{14mu} \frac{\partial w}{\partial y}} = 0}$

For y=0 and y=L_(y):

$w = {{0\mspace{14mu} {and}\mspace{14mu} \frac{\partial w}{\partial x}} = 0}$

Expression of Displacement:

The following expression of displacement w satisfies all the boundaryconditions detailed above:

$\begin{matrix}{{w\left( {x,y} \right)} = {{{{C_{mn}\left\lbrack {1 - {\cos \left( \frac{m \cdot 2 \cdot \pi \cdot x}{L_{x}} \right)}} \right\rbrack}\left\lbrack {1 - {\cos \left( \frac{n \cdot 2 \cdot \pi \cdot y}{L_{y}} \right)}} \right\rbrack}\left( {m,n} \right)} \in N^{2}}} & {{Equation}\mspace{14mu} 6\text{-}28}\end{matrix}$

Buckling flow capacity (N_(x) _(comb) ^(c), N_(y) _(comb) ^(c)):

The expression of N_(x) _(comb) ⁰ according to λ must satisfy:

$N_{x_{comb}}^{0} = {{{\frac{4\pi^{2}}{{L_{y}^{2}m^{2}} + {\lambda \; L_{x}^{2}n^{2}}}\left\lbrack {{{\Omega_{1}\left( \frac{L_{y}}{L_{x}} \right)}^{2}m^{4}} + {{\Omega_{2}\left( \frac{L_{x}}{L_{y}} \right)}^{2}n^{4}} + {\frac{1}{3}\Omega_{3}m^{2}n^{2}}} \right\rbrack}\left( {m,n} \right)} \in N^{2}}$

thus, the buckling flow capacities are obtained:

$\begin{matrix}{{N_{x_{comb}}^{c} = {{Min}\left\{ {{\frac{4\pi^{2}}{{L_{y}^{2}m^{2}} + {\lambda \; L_{x}^{2}n^{2}}}\left\lbrack {{{\Omega_{1}\left( \frac{L_{y}}{L_{x}} \right)}^{2}m^{4}} + {{\Omega_{2}\left( \frac{L_{x}}{L_{y}} \right)}^{2}n^{4}} + {\frac{1}{3}\Omega_{3}m^{2}n^{2}}} \right\rbrack},{\left( {m,n} \right) \in N^{2}}} \right\}}}\mspace{20mu} {N_{y_{comb}}^{c} = {\lambda \; N_{x_{comb}}^{c}}}} & {{Equation}\mspace{14mu} 6\text{-}2}\end{matrix}$

6.4.5 Longitudinal and Shear Compression Flow

The plate is subjected to combined loading: uniform longitudinal(according to the axis X) and shear compression flow: −N_(x) _(comb) ⁰and −N_(y) _(comb) ⁰

Because of this: N_(y) ⁰=p_(z)=0.

Interaction Equation:

The interaction equation for the combined flows of longitudinal andshear compression is:

R _(x) +R _(xy) ^(1.75)=1   Equation 6-30

With

$R_{x} = \frac{N_{x_{comb}}^{c}}{N_{x}^{c}}$

ratio of longitudinal compression flow

$R_{xy} = \frac{N_{{xy}_{comb}}^{c}}{N_{xy}^{c}}$

ratio of shear flow

where N_(x) ^(c) and N_(xy) ^(c) are the buckling flow capacitiescalculated above for a uniaxial loading.

6.4.6 Transversal and Shear Compression Flow

The plate is subjected to a uniform transversal compression flow(according to the y axis) and a shear flow: −N_(y) _(comb) ⁰ and −N_(xy)_(comb) ⁰

Because of this: N_(x) ⁰=p_(z)=0.

The interaction equation for the combined flows of transversal and shearcompression is:

R _(y) +R _(xy) ^(1.75)=1   Equation 6-31

With:

$R_{y} = \frac{N_{y_{comb}}^{c}}{N_{y}^{c}}$

ratio of transversal compression flow

$R_{xy} = \frac{N_{{xy}_{comb}}^{c}}{N_{xy}^{c}}$

ratio of shear flow

where N_(y) ^(c) and N_(xy) ^(c) are the buckling capacity flowscalculated above for a uniaxial loading.

6.4.7 Biaxial and Shear Compression Flow

The plate is subjected to combined loadings: a uniform longitudinalcompression flow (according to the x axis) and a uniform transversalcompression flow (according to the y axis) as well as a buckling flow:−N_(x) _(comb) ⁰; −N_(y) _(comb) ⁰ and −N_(xy) _(comb) ⁰.

Because of this: p_(z)=0.

The interaction equation is obtained in two steps. Firstly, we determinea reserve factor RF_(bi) corresponding to the biaxial compression flow:

$\begin{matrix}{{RF}_{bi} = {\frac{N_{x_{comb}}^{c}}{N_{x_{comb}}^{0}} = \frac{N_{y_{comb}}^{c}}{N_{y_{comb}}^{0}}}} & {{Equation}\mspace{14mu} 6\text{-}32}\end{matrix}$

Then this value is used in an interaction equation for the combinedflows of biaxial and shear compression:

R _(bi) +R _(xy) ^(1.75)=1   Equation 6-33

With:

-   -   R_(bi)=RF_(bi) ⁻¹ ratio of biaxial compression flow

$R_{xy} = \frac{N_{{xy}_{comb}}^{0}}{N_{xy}^{c}}$

ratio of shear flow

where N_(xy) ^(c) is the buckling flow capacity calculated in pure shearload.

The method according to the invention also includes an iteration loop(see FIG. 7). This loop makes it possible to modify the value of appliedloads, or the dimensional values of panels stiffened by triangularpockets in consideration, according to the results of at least one ofsteps 3 to 6.

The method, such as has been described, can be implemented at leastpartially in the form of a macro on a spreadsheet type programme.

Such a programme used thus, for example, for entering material andgeometry data stored in a dedicated zone, as well as various cases ofconsidered loads and boundary conditions, and supplying exit values forpanel mass, reserve factor at ultimate load concerning in particulartriangular pockets, stiffeners and general failure. These exit data thushighlight the cases of loads or dimensioning which are incompatible withthe desired reserve factors.

ADVANTAGES OF THE INVENTION

We understand that the NASA process previously known, has beensubstantially extended in the frame of the present invention to takeinto account the particularities of the aeronautical domain:

-   -   Local capacity values for stiffeners (destruction, lateral        instability etc.) for compression according to direction X or Y        and shear load,    -   Local capacity values of the triangular skin for compression        according to direction X or Y and shear load,    -   Plasticity correction,    -   Preliminary mass calculation,        -   Calculation of general buckling for a compression according            to direction X or Y and a shear load.

The principal improvements are the calculation of stress capacities forthe different types of buckling and the calculations of adapted reservefactors.

Extensions: Case of Loading

-   -   double-compression (for local and global buckling)    -   Combined loading: compression and shear load

Extensions: Improvements of the Parameters of the Method

-   -   No limitation on the material's Poisson coefficient    -   Variation of the grid angle (different by 60°)        -   Plasticity where the structure stiffened by the triangular            pockets is considered as an equivalent stiffened panel        -   Boundary conditions (clamping or intermediate boundary            conditions) on the local or global buckling One of the most            significant advantages of the method of dimensioning            according to the invention is the possibility of installing            panels stiffened by triangular pockets, instead of and in            place of the panels previously created with two            perpendicular families of stiffeners (longerons and ribs)            underneath, resulting, for equal mechanical resistance, in a            mass gain reaching 30% on some pieces.

VARIATIONS OF THE INVENTION

The scope of the present invention is not limited to details of thetypes of embodiment considered above as an example, but extends on thecontrary to modifications to the scope of those skilled in the art.

In the present description we have referred to isosceles triangle baseangles of between 45° and 70°, which correspond to current requirementsfor aeronautical structures. It is however clear that a similar methodcan be implemented for all isosceles angle values in panels stiffened bytriangular pockets.

1. (canceled) 2: A method for dimensioning of a substantially planepanel of homogeneous and isotropic material by an analytical procedure,the panel having a skin reinforced by an assembly of three parallelbundles of stiffeners integrated with the panel, and triangular pocketsdefined on the skin by the stiffeners, wherein the stiffeners arestrip-shaped and the panel must satisfy a specification of mechanicalresistance to predetermined external loads, the method includes:acquiring values of input data commensurate with a geometry of thepanel, a material of the panel, and loading applied to the panel;calculating stresses applied on the skin of the panel and thestiffeners, based on the geometry of the panel stiffened by thetriangular pockets, and on applied external loads; calculating internalloads induced on the panel; performing a strength analysis includingcalculating reserve factors of the material at a limit load and at anultimate load; calculating local stresses admissible by the panel,including calculating the reserve factors at maximum stresses; andcalculating general instability of the panel, including calculating areserve factor for a stiffened flat panel under pure or combined loadingconditions. 3: The method of claim 2, wherein the input data includesthe mechanical parameters related to the material, dimensions of thepanel, cross sections of the stiffeners, dimensions of the core, aconstant thickness of the panel and limit loads of the panel. 4: Themethod of claim 2, wherein calculating the internal loads induced on thepanel further include performing a correction of the applied loads thattakes plasticity into account using an iterative method of calculationof plastic stresses, carried out until five parameters of the material(E_(0°) ^(st), E_(+□) ^(st), E_(−□) ^(st), E_(skin), □_(ep)) acquiredduring the first step are substantially equal to the same parametersobtained after calculation of plastic stress. 5: The method of claim 2,wherein calculating the local stresses further includes calculatingadmissible buckling flows and reserve factors for isosceles triangularpockets, wherein the applied stresses taken into account for calculatingthe reserve factor are stresses acting exclusively in the skin andexternal flows used are skin flows that do not correspond to a completeloading of the panel. 6: The method of claim 5, wherein calculatingadmissible buckling flows further includes calculating admissible valuesfor plates subjected to cases of pure loading (compression in twodirections in the plane, shear) by using a finite elements procedure,and calculating curves of interaction between these cases of pureloading. 7: The method of claim 6, wherein calculating admissible valuesincludes creating a parametric finite elements model of a triangularplate, executing numerous different combinations to obtain bucklingresults, obtaining parameters compatible with a polynomial analyticalformulation. 8: The method of claim 7, wherein, in the case of pureloading, the interaction curves are defined by creating finite elementsmodels of a plurality of triangular plates having different isoscelesangles, wherein the isosceles angle θ is defined as a base angle of theisosceles triangles, and for each isosceles angle the following stepsare performed: determining the admissible folding flow (without plasticcorrection) for diverse plate thicknesses via finite elements model,plotting a curve of admissible buckling flow as a function of the ratio$\frac{D}{h^{2}}\left( D \right.$ plate stiffness, h height of thetriangle), wherein this curve is determined for small values of theratio $\frac{D}{h^{2}}$ by a second degree equation that is a functionof this ratio, in which the coefficients K₁ and K₂ depend on the angleand on the load case under consideration, and plotting the evolution ofthe coefficients K₁ and K₂ of the polynomial equation as a function ofthe base angle of the isosceles triangle, wherein these coefficients areplotted as a function of the angle of the triangular plates underconsideration, then interpolating to determine a polynomial equationwith which these constants can be calculated regardless of the isoscelesangle. 9: The method of claim 7, wherein in the case of combinedloading, the following hypothesis is used: if certain components of thecombined load are in tension, then these components are not taken intoaccount for the calculation, and in that the interaction curves aredefined by creating finite elements models of a plurality of triangularplates having different isosceles angles, wherein the isosceles angle isdefined as the base angle of the isosceles triangle, the following stepare performed determining the inherent buckling value corresponding todifferent distributions of external loads, plotting interaction curvesfor each angle and each combination of loads, then approximating thesecurves with a single equation covering all combinations:${{E_{eX}^{A} + R_{cY}^{B} + R_{s}^{C}} = {1\mspace{11mu} \left( {{{{where}\mspace{14mu} R_{i}} = \frac{N_{i}^{app}}{N_{i}^{crit}}},{i = {cX}},{{cY}\mspace{14mu} {or}\mspace{14mu} s}} \right)}},$wherein A, B, C are empirical coefficients. 10: The method of one ofclaim 7, wherein in the case of isosceles triangular plates in simple ornested bracing relationship, in case of combined loading, oneinteraction curve: R_(cX)+R_(cY)+R_(s) ^(3/2)=1, is used for all loadingcases. 11: The method of claim 10, wherein calculating the reservefactor is calculated on the basis of external flows of the stiffenedpanel and of limit conditions in simple or nested bracing relationship.12: A non-transitory computer readable medium storing computer-readableinstructions therein which when executed by a computer cause thecomputer to perform a method for dimensioning of a substantially planepanel, the method comprising: acquiring values of input datacommensurate with a geometry of the panel, a material of the panel, andloading applied to the panel; calculating stresses applied on the skinof the panel and the stiffeners, based on the geometry of the panelstiffened by the triangular pockets, and on applied external loads;calculating internal loads induced on the panel; performing a strengthanalysis including calculating reserve factors of the material at alimit load and at an ultimate load; calculating local stressesadmissible by the panel, including calculating the reserve factors atmaximum stresses; and calculating general instability of the panel,including calculating a reserve factor for a stiffened flat panel underpure or combined loading conditions.